Daily Papers

We respond to the recent paper by Makelov et al. (2023), which reviews subspace interchange intervention methods like distributed alignment search (DAS; Geiger et al. 2023) and claims that these methods potentially cause "interpretability illusions". We first review Makelov et al. (2023)'s technical notion of what an "interpretability illusion" is, and then we show that even intuitive and desirable explanations can qualify as illusions in this sense. As a result, their method of discovering "illusions" can reject explanations they consider "non-illusory". We then argue that the illusions Makelov et al. (2023) see in practice are artifacts of their training and evaluation paradigms. We close by emphasizing that, though we disagree with their core characterization, Makelov et al. (2023)'s examples and discussion have undoubtedly pushed the field of interpretability forward.

We present a method for diagnosing interpretation in neural networks by identifying an input subspace where a proposed interpretation is highly faithful. Our method is particularly useful for causal-abstraction-style interpretability, where a high-level causal hypothesis is evaluated by interchange interventions. Rather than treating interchange intervention accuracy as a single global summary, we refine this framework by partitioning the input space into well-interpreted and under-interpreted regions according to pairwise interchange-intervention behavior. This turns causal abstraction from a purely global evaluation into a more diagnostic tool: it not only measures whether an interpretation works, but also reveals where it works, where it fails, and what distinguishes the two cases. This diagnostic view also provides practical heuristics for improving interpretations. By analyzing the structure of the well-interpreted and under-interpreted regions, we can identify missing distinctions in a high-level hypothesis, discover previously unmodeled intermediate variables, and combine complementary partial interpretations into a stronger one. We instantiate this idea as a simple four-step recipe and show that it yields informative error analyses across multiple causal abstraction settings. In a toy logic task, recursively applying the recipe recovers a high-level hypothesis from scratch. More broadly, our results suggest that partitioning the input space is a useful step toward more precise, constructive, and scalable mechanistic interpretability.

Mechanistic interpretability aims to understand model behaviors in terms of specific, interpretable features, often hypothesized to manifest as low-dimensional subspaces of activations. Specifically, recent studies have explored subspace interventions (such as activation patching) as a way to simultaneously manipulate model behavior and attribute the features behind it to given subspaces. In this work, we demonstrate that these two aims diverge, potentially leading to an illusory sense of interpretability. Counterintuitively, even if a subspace intervention makes the model's output behave as if the value of a feature was changed, this effect may be achieved by activating a dormant parallel pathway leveraging another subspace that is causally disconnected from model outputs. We demonstrate this phenomenon in a distilled mathematical example, in two real-world domains (the indirect object identification task and factual recall), and present evidence for its prevalence in practice. In the context of factual recall, we further show a link to rank-1 fact editing, providing a mechanistic explanation for previous work observing an inconsistency between fact editing performance and fact localization. However, this does not imply that activation patching of subspaces is intrinsically unfit for interpretability. To contextualize our findings, we also show what a success case looks like in a task (indirect object identification) where prior manual circuit analysis informs an understanding of the location of a feature. We explore the additional evidence needed to argue that a patched subspace is faithful.

Low-Rank Adaptation (LoRA) merging can efficiently combine diverse generative capabilities from multiple trained LoRAs for a diffusion model. However, existing LoRA merging techniques often suffer from severe parameter interference, causing destructive collisions in the shared parameter space. To address this, we propose Subspace Signal Routing (SSR), which resolves interference by routing internal signals instead of performing parameter-space merge. Specifically, SSR first constructs a unified subspace by concatenating candidate LoRAs along the rank dimension. Next, SSR employs an inverse correlation matrix to decorrelate mixed signals within this space. Finally, a directional guide matrix steers these purified signals into their respective task-specific subspaces. We provide a rigorous theoretical analysis proving that SSR aligns with the Ordinary Least Squares (OLS) solution, thereby ensuring mathematical optimality. We utilize the additivity of sufficient statistics to design a streaming algorithm. This enables on-the-fly updates that significantly reduce memory overhead and computation time. Extensive experiments validate that SSR significantly outperforms state-of-the-art methods while maintaining comparable efficiency. Code is available at https://github.com/nagara214/SSR-Merge.

Mechanistic interpretability seeks to understand the internal mechanisms of machine learning models, where localization -- identifying the important model components -- is a key step. Activation patching, also known as causal tracing or interchange intervention, is a standard technique for this task (Vig et al., 2020), but the literature contains many variants with little consensus on the choice of hyperparameters or methodology. In this work, we systematically examine the impact of methodological details in activation patching, including evaluation metrics and corruption methods. In several settings of localization and circuit discovery in language models, we find that varying these hyperparameters could lead to disparate interpretability results. Backed by empirical observations, we give conceptual arguments for why certain metrics or methods may be preferred. Finally, we provide recommendations for the best practices of activation patching going forwards.

Arditi et al. (2024) has shown that refusal in safety fine-tuned chat models is mediated by a single linear direction in the residual stream, recoverable by a difference-in-means (DiM) of harmful and harmless activations. We compare DiM-based interventions (activation addition and directional ablation) with two interventions derived from Iterative Nullspace Projection (INLP) -- nullspace projection and counterfactual flipping -- on five open-weight chat models, asking whether INLP can match DiM at steering refusal and whether its richer parameterisation yields more tweakable interventions. INLP counterfactual flipping is competitive with DiM directional ablation on refusal suppression, while nullspace projection is consistently weaker. Restricting INLP to the leading directions of the extracted subspace preserves most of the suppression effect at near-baseline perplexity, giving a tunable capability. Geometrically, the two INLP interventions land in qualitatively different regions of activation space: nullspace projection collapses transformed activations between the harmful and harmless clusters, while counterfactual flipping moves them into the opposite cluster, suggesting that the model encodes the absence of a concept differently from its opposite -- an intriguing distinction that warrants further investigation in future work.

Intervention-based model steering offers a lightweight and interpretable alternative to prompting and fine-tuning. However, by adapting strong optimization objectives from fine-tuning, current methods are susceptible to overfitting and often underperform, sometimes generating unnatural outputs. We hypothesize that this is because effective steering requires the faithful identification of internal model mechanisms, not the enforcement of external preferences. To this end, we build on the principles of distributed alignment search (DAS), the standard for causal variable localization, to propose a new steering method: Concept DAS (CDAS). While we adopt the core mechanism of DAS, distributed interchange intervention (DII), we introduce a novel distribution matching objective tailored for the steering task by aligning intervened output distributions with counterfactual distributions. CDAS differs from prior work in two main ways: first, it learns interventions via weak-supervised distribution matching rather than probability maximization; second, it uses DIIs that naturally enable bi-directional steering and allow steering factors to be derived from data, reducing the effort required for hyperparameter tuning and resulting in more faithful and stable control. On AxBench, a large-scale model steering benchmark, we show that CDAS does not always outperform preference-optimization methods but may benefit more from increased model scale. In two safety-related case studies, overriding refusal behaviors of safety-aligned models and neutralizing a chain-of-thought backdoor, CDAS achieves systematic steering while maintaining general model utility. These results indicate that CDAS is complementary to preference-optimization approaches and conditionally constitutes a robust approach to intervention-based model steering. Our code is available at https://github.com/colored-dye/concept_das.

Large Language Models (LLMs) are powerful text generators, yet they can produce toxic or harmful content even when given seemingly harmless prompts. This presents a serious safety challenge and can cause real-world harm. Toxicity is often subtle and context-dependent, making it difficult to detect at the token level or through coarse sentence-level signals. Moreover, efforts to mitigate toxicity often face a trade-off between safety and the coherence, or fluency of the generated text. In this work, we present a targeted subspace intervention strategy for identifying and suppressing hidden toxic patterns from underlying model representations, while preserving overall ability to generate safe fluent content. On the RealToxicityPrompts, our method achieves strong mitigation performance compared to existing baselines, with minimal impact on inference complexity. Across multiple LLMs, our approach reduces toxicity of state-of-the-art detoxification systems by 8-20%, while maintaining comparable fluency. Through extensive quantitative and qualitative analyses, we show that our approach achieves effective toxicity reduction without impairing generative performance, consistently outperforming existing baselines.

Large Language Models (LLMs) rely on safety alignment to produce socially acceptable responses. This is typically achieved through instruction tuning and reinforcement learning from human feedback. However, this alignment is known to be brittle: further fine-tuning, even on benign or lightly contaminated data, can degrade safety and reintroduce harmful behaviors. A growing body of work suggests that alignment may correspond to identifiable geometric directions in weight space, forming subspaces that could, in principle, be isolated or preserved to defend against misalignment. In this work, we conduct a comprehensive empirical study of this geometric perspective. We examine whether safety-relevant behavior is concentrated in specific subspaces, whether it can be separated from general-purpose learning, and whether harmfulness arises from distinguishable patterns in internal representations. Across both parameter and activation space, our findings are consistent: subspaces that amplify safe behaviors also amplify unsafe ones, and prompts with different safety implications activate overlapping representations. We find no evidence of a subspace that selectively governs safety. These results challenge the assumption that alignment is geometrically localized. Rather than residing in distinct directions, safety appears to emerge from entangled, high-impact components of the model's broader learning dynamics. This suggests that subspace-based defenses may face fundamental limitations and underscores the need for alternative strategies to preserve alignment under continued training. We corroborate these findings through multiple experiments on five open-source LLMs. Our code is publicly available at: https://github.com/CERT-Lab/safety-subspaces.

Model merging aims to enable multi-task learning by integrating the capabilities of multiple models fine-tuned from the same pre-trained checkpoint into a single model. Its core challenge is inter-task interference among task-specific parameter updates. In this paper, we analyze the output shifts induced by task updates and observe that their energy is concentrated in a small number of principal directions. We call the subspace spanned by these directions the essential subspace. In contrast, most remaining directions carry little task-relevant energy, but their accumulation across multiple task updates can cause severe interference during merging. Motivated by this observation, we propose Essential Subspace Decomposition (ESD), which decomposes each task update according to the principal components of its activation shift. Based on ESD, we introduce Essential Subspace Merging (ESM), a training-free static merging method that orthogonalizes and fuses essential components into one compact multi-task model. We further extend ESM to ESM++, a training-free dynamic merging method that decomposes task-specific residuals into low-rank experts and selects the most relevant expert through prototype-based routing during forward inference. Extensive experiments across multiple task sets and model scales demonstrate that ESM and ESM++ effectively preserves task knowledge while reducing inter-task interference.

Merging models fine-tuned from a common, extensively pre-trained large model but specialized for different tasks has been demonstrated as a cheap and scalable strategy to construct a multi-task model that performs well across diverse tasks. Recent research, exemplified by task arithmetic, highlights that this multi-task model can be derived through arithmetic operations on task vectors. Nevertheless, current merging techniques frequently resolve potential conflicts among parameters from task-specific models by evaluating individual attributes, such as the parameters' magnitude or sign, overlooking their collective impact on the overall functionality of the model. In this work, we propose the CONtinuous relaxation of disCRETE (Concrete) subspace learning method to identify a common low-dimensional subspace and utilize its shared information to track the interference problem without sacrificing much performance. Specifically, we model the problem as a bi-level optimization problem and introduce a meta-learning framework to find the Concrete subspace mask through gradient-based techniques. At the upper level, we focus on learning a shared Concrete mask to identify the subspace, while at the inner level, model merging is performed to maximize the performance of the merged model. We conduct extensive experiments on both vision domain and language domain, and the results demonstrate the effectiveness of our method. The code is available at https://github.com/tanganke/subspace_fusion

We tackle the problem disentangling the latent space of an autoencoder in order to separate labelled attribute information from other characteristic information. This then allows us to change selected attributes while preserving other information. Our method, matrix subspace projection, is much simpler than previous approaches to latent space factorisation, for example not requiring multiple discriminators or a careful weighting among their loss functions. Furthermore our new model can be applied to autoencoders as a plugin, and works across diverse domains such as images or text. We demonstrate the utility of our method for attribute manipulation in autoencoders trained across varied domains, using both human evaluation and automated methods. The quality of generation of our new model (e.g. reconstruction, conditional generation) is highly competitive to a number of strong baselines.

Large language models (LLMs) handle many tasks with one set of parameters, but under KV-cached inference it is unclear what task-general structure, if any, is used at decode time rather than during prefill. We propose DecodeShare, a protocol that identifies a low-dimensional subspace consistently shared across tasks in decode-time hidden states, and then tests its causal role by removing that subspace only during decoding. In our experiments, disturbing the discovered shared subspace degrades decision performance far more than disturbing either a prefill-derived or random subspace under the same intervention budget. We further show this decode-shared subspace has practical consequences for activation steering: common steering directions can overlap the task-general decode channel. Projecting out this shared subspace directly separates the functional roles of the two components, while evaluating steering vectors at decode-time yields more reliable signal for downstream deployment than prefill-based proxies. Despite its compactness, the shared subspace can serve as a high-leverage causal channel at decode time. Code is available at: https://github.com/Zishan-Shao/decodeshare.git.

Vision-Language Models require efficient adaptation to continually emerging downstream tasks. While Parameter-Efficient Fine-Tuning mitigates catastrophic forgetting, assigning isolated modules per task leads to parameter explosion. Conversely, recent similarity-driven sharing mechanisms falsely equate superficial visual similarity with underlying alignment consistency. This fundamental mismatch triggers severe negative transfer between visually similar but logically distinct tasks and fails to exploit alignment reuse across visually diverse ones. We argue thatalignment sharing is fundamentally a geometric problem of overlapping optimization trajectories within shared low-rank subspaces. Grounded in this insight, we propose iGSP, a novel framework that achieves efficient adaptation via implicit gradient subspace projection. Leveraging the early convergence of MoE routers to establish the subspace basis, iGSP bifurcates the adaptation process into two phases. First, the Subspace Identification phase introduces candidate experts via basis pre-expansion, applies a novel subspace-constrained regularization to implicitly project new task gradients onto the historical subspace, and precisely prunes redundant dimensions by treating routing probabilities as gradient flow indicators, ultimately to maximize knowledge reuse. Second, the Orthogonal Subspace Fine-Tuning phase fixes this structural basis and removes the regularization to rapidly fit the task-specific residual loss. Extensive experiments on the MTIL benchmark demonstrate that iGSP achieves state-of-the-art accuracy while significantly improving training efficiency, reducing the average trainable parameters by 42.7\% compared to current SOTA methods, and decreasing the final total parameters by 86.9\% relative to counterparts. The source code is available at https://github.com/GeoX-Lab/iGSP.

Detecting visual anomalies in industrial inspection often requires training with only a few normal images per category. Recent few-shot methods achieve strong results employing foundation-model features, but typically rely on memory banks, auxiliary datasets, or multi-modal tuning of vision-language models. We therefore question whether such complexity is necessary given the feature representations of vision foundation models. To answer this question, we introduce SubspaceAD, a training-free method, that operates in two simple stages. First, patch-level features are extracted from a small set of normal images by a frozen DINOv2 backbone. Second, a Principal Component Analysis (PCA) model is fit to these features to estimate the low-dimensional subspace of normal variations. At inference, anomalies are detected via the reconstruction residual with respect to this subspace, producing interpretable and statistically grounded anomaly scores. Despite its simplicity, SubspaceAD achieves state-of-the-art performance across one-shot and few-shot settings without training, prompt tuning, or memory banks. In the one-shot anomaly detection setting, SubspaceAD achieves image-level and pixel-level AUROC of 97.1% and 97.5% on the MVTec-AD dataset, and 93.4% and 98.2% on the VisA dataset, respectively, surpassing prior state-of-the-art results. Code and demo are available at https://github.com/CLendering/SubspaceAD.

Representation intervention has emerged as a promising paradigm for aligning large language models toward desired behaviors without modifying model weights. Existing methods typically apply a fixed intervention uniformly across all inputs. However, we find that the appropriate intervention direction and strength vary substantially across samples, and such indiscriminate intervention leads to degradation of general capabilities on benign inputs. To address these challenges, we propose Multi-Adapter Representation Interventions via Energy Calibration (MARI). Specifically, we introduce a competitive multi-adapter mechanism in which specialized experts capture non-linear correction patterns and adaptively determine the appropriate intervention direction and strength for different samples. Furthermore, we design an energy-based gating module that leverages internal propagation dynamics to distinguish inputs that are applicable for intervention. Extensive experiments across diverse model families and parameter scales demonstrate that MARI achieves state-of-the-art alignment performance. Our method significantly improves performance on TruthfulQA, BBQ, and safety benchmarks, while maintaining and even improving general capabilities on tasks such as MMLU and ARC. Our code is available at https://github.com/V1centNevwake/MARI.

Traditional white-box methods for creating adversarial perturbations against LLMs typically rely only on gradient computation from the targeted model, ignoring the internal mechanisms responsible for attack success or failure. Conversely, interpretability studies that analyze these internal mechanisms lack practical applications beyond runtime interventions. We bridge this gap by introducing a novel white-box approach that leverages mechanistic interpretability techniques to craft practical adversarial inputs. Specifically, we first identify acceptance subspaces - sets of feature vectors that do not trigger the model's refusal mechanisms - then use gradient-based optimization to reroute embeddings from refusal subspaces to acceptance subspaces, effectively achieving jailbreaks. This targeted approach significantly reduces computation cost, achieving attack success rates of 80-95\% on state-of-the-art models including Gemma2, Llama3.2, and Qwen2.5 within minutes or even seconds, compared to existing techniques that often fail or require hours of computation. We believe this approach opens a new direction for both attack research and defense development. Furthermore, it showcases a practical application of mechanistic interpretability where other methods are less efficient, which highlights its utility. The code and generated datasets are available at https://github.com/Sckathach/subspace-rerouting.

Safety-aligned LLMs can exhibit emergent misalignment (EM): narrow domain adaptation unexpectedly triggers catastrophic safety failures across unrelated domains. Prior static analyses leave training dynamics unmapped, while existing defenses rely on heuristics that degrade utility. We present a dynamic, second-order geometric study of EM. Tracking training trajectories reveals that directional Hessian curvature concentrates sharply on semantic pivot tokens. Grassmannian projections show that, in most settings, harmful-safe gap widens mainly because safe-gradient overlap declines. Leveraging these insights, we introduce a parameter-level Geometric Mitigation Framework that orthogonally projects empirical harmful gradient subspace out of parameter updates. On Qwen2.5-14B-IT, our defense suppresses free-generation EM by up to 80.0%; across the other three of four open-weight instruction-based model families (3B--20B), where single-layer behavioral EM is already near zero, teacher-forced evaluation shows same harmful subspace controls the conditional support of frozen EM responses. Crucially, these diagnostics unmask the illusion of behavioral safety: the same subspace remains measurable and steerable in models where behavioral EM is near zero. Code: https://github.com/WeiqiaoQUE/mechanistic-emergent-misalignment.

While orthogonal subspace methods try to mitigate task interference in Continual Learning (CL), they often suffer from energy diffusion across the basis, hindering knowledge compaction and exhausting capacity for future tasks. We observe that output feature drift induced by parameter updates is inherently low-rank, and theoretically prove that preserving parameters along the principal directions of this drift minimizes the output reconstruction error. Motivated by this, we propose Energy-Concentrated and Energy-Ordered Low-Rank Adaptation (E^2-LoRA). By explicitly ordering and concentrating knowledge into leading ranks, E^2-LoRA frees capacity for subsequent tasks. Furthermore, we design a dynamic rank allocation strategy to balance stability and plasticity by jointly optimizing energy retention and model plasticity. Extensive experiments across multiple benchmarks demonstrate that E^2-LoRA achieves state-of-the-art performance.

Mixture-of-Experts (MoE) architectures achieve parameter efficiency through conditional computation, yet contemporary designs suffer from two fundamental limitations: structural parameter isolation that causes catastrophic forgetting, and instruction-overfitting that degrades performance in instruction-free scenarios. We propose CDSP-MoE (Conflict-Driven Subspace Pruning MoE), a framework that addresses these issues through a paradigm shift from isolated expert containers to dynamic expert instantiation within a shared physical subspace. Grounded in the Universal Weight Subspace Hypothesis, CDSP-MoE maintains a super-complete parameter backbone where logical experts are carved out via learnable topology masks. Unlike prior work that uses gradient conflict for token reassignment or optimization surgery, we leverage it as a structural supervisory signal: a Lagged Gradient Game penalizes interfering connections in the shared manifold, enabling the topology to spontaneously prune conflicting pathways and evolve interpretable modular structures. Experimental results demonstrate that CDSP-MoE achieves robust content-driven routing without human-defined task labels, maintaining semantic specialization even under strict blind inference protocols where explicit instructions are absent. Code is available at: https://github.com/konodiodaaaaa1/Conflict-Driven-Subspace-Pruning-Mixture-of-Experts

Model merging aims to integrate multiple task-specific fine-tuned models derived from a shared pre-trained checkpoint into a single multi-task model without additional training. Despite extensive research, task interference remains a major obstacle that often undermines the performance of merged models. In this paper, we propose ESM (Essential Subspace Merging) , a robust framework for effective model merging. We begin by performing Principal Component Analysis (PCA) on feature shifts induced by parameter updates. The resulting principal directions span an essential subspace that dominantly influences feature representations. Each task's parameter update matrix is projected onto its respective essential subspace for low-rank decomposition before merging. This methodology mitigates inter-task interference while preserving core task-specific functionality. Furthermore, we introduce a multi-level polarized scaling strategy that amplifies parameters containing critical knowledge and suppresses redundant ones, preventing essential knowledge from being overwhelmed during fusion. Extensive experiments across multiple task sets and model scales demonstrate that our method achieves state-of-the-art performance in multi-task model merging.

When researchers ask whether two transformer layers are "equivalent" for compression, they often conflate distinct tests. Replacement asks whether one layer's map can substitute for another's in place; interchange asks whether two layers approximately commute when their positions are swapped. Both are output-grounded swap-KL probes, but they need not agree: on pretrained transformers the protocol gap can change which layers look safe to prune by several-fold under the same evaluator, especially when replacement distances are high. We measure both protocols across checkpoints and architectures. On a Pythia training trajectory (410M and 1.4B), the replacement-interchange gap grows from initialization to convergence. Under one matched WikiText-2 contract at 8B scale, Qwen3-8B enters a divergent regime: interchange-guided removal is several-fold safer than replacement-guided at the same layer budgets, while Llama-3.1-8B ties the two protocols for pruning cost even though interchange KL is lower, showing metric gaps need not map one-to-one to removal. Before layer removal or merging, score both swap-KLs on the target checkpoint; the diagnostic requires only unlabeled forward passes.

Policy gradient methods hold great potential for solving complex continuous control tasks. Still, their training efficiency can be improved by exploiting structure within the optimization problem. Recent work indicates that supervised learning can be accelerated by leveraging the fact that gradients lie in a low-dimensional and slowly-changing subspace. In this paper, we conduct a thorough evaluation of this phenomenon for two popular deep policy gradient methods on various simulated benchmark tasks. Our results demonstrate the existence of such gradient subspaces despite the continuously changing data distribution inherent to reinforcement learning. These findings reveal promising directions for future work on more efficient reinforcement learning, e.g., through improving parameter-space exploration or enabling second-order optimization.

Distributed training of foundation models via DDP is limited by interconnect bandwidth. While infrequent communication strategies reduce synchronization frequency, they remain bottlenecked by the memory and communication requirements of optimizer states. Low-rank optimizers can alleviate these constraints; however, in the local-update regime, workers lack access to the full-batch gradients required to compute low-rank projections, which degrades performance. We propose LoRDO, a principled framework unifying low-rank optimization with infrequent synchronization. We first demonstrate that, while global projections based on pseudo-gradients are theoretically superior, they permanently restrict the optimization trajectory to a low-rank subspace. To restore subspace exploration, we introduce a full-rank quasi-hyperbolic update. LoRDO achieves near-parity with low-rank DDP in language modeling and downstream tasks at model scales of 125M--720M, while reducing communication by approx 10 times. Finally, we show that LoRDO improves performance even more in very low-memory settings with small rank/batch size.

Continual learning aims to learn multiple tasks sequentially while preserving prior knowledge, but faces the challenge of catastrophic forgetting when adapting to new tasks. Recently, approaches leveraging pre-trained models have gained increasing popularity in mitigating this issue, due to the strong generalization ability of foundation models. To adjust pre-trained models for new tasks, existing methods usually employ low-rank adaptation, which restricts parameter updates to a fixed low-rank subspace. However, constraining the optimization space inherently compromises the model's learning capacity, resulting in inferior performance. To address this limitation, we propose Continuous Subspace Optimization for Continual Learning (CoSO) to fine-tune the model in a series of subspaces rather than a single one. These sequential subspaces are dynamically determined through the singular value decomposition of the gradients. CoSO updates the model by projecting gradients onto these subspaces, ensuring memory-efficient optimization. To mitigate forgetting, the optimization subspace of each task is constrained to be orthogonal to the historical task subspace. During task learning, CoSO maintains a task-specific component that captures the critical update directions for the current task. Upon completing a task, this component is used to update the historical task subspace, laying the groundwork for subsequent learning. Extensive experiments on multiple datasets demonstrate that CoSO significantly outperforms state-of-the-art methods, especially in challenging scenarios with long task sequences.

Mixture of Experts (MoE) models scale capacity but often suffer from representation collapse and gradient instability. We propose Dynamic Subspace Composition (DSC), a framework that approximates context-dependent weights via a state-dependent, sparse expansion of a shared basis bank. Formally, DSC models the weight update as a residual trajectory within a Star- Shaped Domain, employing a Magnitude-Gated Simplex Interpolation to ensure continuity at the identity. Unlike standard Mixture-of-LoRAs, which incurs O(M rd) parameter complexity by retrieving independent rank-r matrices, DSC constructs a compositional rank-K approximation from decoupled unit-norm basis vectors. This reduces parameter complexity to O(M d) and memory traffic to O(Kd), while Frame-Theoretic regularization and spectral constraints provide rigorous worst-case bounds on the dynamic update. The code is available at https://github. com/VladimerKhasia/DSC

Targeted data selection has emerged as a crucial paradigm for efficient instruction tuning, aiming to identify a small yet influential subset of training examples for a specific target task. In practice, influence is often measured through the effect of an example on parameter updates. To make selection scalable, many approaches leverage optimizer statistics (e.g., Adam states) as an axis-aligned surrogate for update geometry (i.e., diagonal precondition), implicitly treating parameters as coordinate-wise independent. We show that this assumption breaks down in parameter-efficient fine-tuning (PEFT) methods such as LoRA. In this setting, the induced optimization geometry exhibits strong cross-parameter coupling with non-trivial off-diagonal interactions, while the task-relevant update directions are confined to a low-dimensional subspace. Motivated by this mismatch, we propose GIST (Gradient Isometric Subspace Transformation), a simple yet principled alternative that replaces axis-aligned scaling with robust subspace alignment. GIST recovers a task-specific subspace from validation gradients via spectral filtering (SVD), projects training gradients into this coupled subspace, and scores examples by their alignment with target directions.Extensive experiments have demonstrated that GIST matches or outperforms the state-of-the-art baseline with only 0.29% of the storage and 25% of the computational time under the same selection budget.

Learning similarity is a key aspect in medical image analysis, particularly in recommendation systems or in uncovering the interpretation of anatomical data in images. Most existing methods learn such similarities in the embedding space over image sets using a single metric learner. Images, however, have a variety of object attributes such as color, shape, or artifacts. Encoding such attributes using a single metric learner is inadequate and may fail to generalize. Instead, multiple learners could focus on separate aspects of these attributes in subspaces of an overarching embedding. This, however, implies the number of learners to be found empirically for each new dataset. This work, Dynamic Subspace Learners, proposes to dynamically exploit multiple learners by removing the need of knowing apriori the number of learners and aggregating new subspace learners during training. Furthermore, the visual interpretability of such subspace learning is enforced by integrating an attention module into our method. This integrated attention mechanism provides a visual insight of discriminative image features that contribute to the clustering of image sets and a visual explanation of the embedding features. The benefits of our attention-based dynamic subspace learners are evaluated in the application of image clustering, image retrieval, and weakly supervised segmentation. Our method achieves competitive results with the performances of multiple learners baselines and significantly outperforms the classification network in terms of clustering and retrieval scores on three different public benchmark datasets. Moreover, our attention maps offer a proxy-labels, which improves the segmentation accuracy up to 15% in Dice scores when compared to state-of-the-art interpretation techniques.

Class-Incremental Learning (CIL) requires a learning system to continually learn new classes without forgetting. Despite the strong performance of Pre-Trained Models (PTMs) in CIL, a critical issue persists: learning new classes often results in the overwriting of old ones. Excessive modification of the network causes forgetting, while minimal adjustments lead to an inadequate fit for new classes. As a result, it is desired to figure out a way of efficient model updating without harming former knowledge. In this paper, we propose ExpAndable Subspace Ensemble (EASE) for PTM-based CIL. To enable model updating without conflict, we train a distinct lightweight adapter module for each new task, aiming to create task-specific subspaces. These adapters span a high-dimensional feature space, enabling joint decision-making across multiple subspaces. As data evolves, the expanding subspaces render the old class classifiers incompatible with new-stage spaces. Correspondingly, we design a semantic-guided prototype complement strategy that synthesizes old classes' new features without using any old class instance. Extensive experiments on seven benchmark datasets verify EASE's state-of-the-art performance. Code is available at: https://github.com/sun-hailong/CVPR24-Ease

The rapid growth of large models has raised concerns about their environmental impact and equity in accessibility due to significant computational costs. Low-Rank Adapters (LoRA) offer a lightweight solution for finetuning large models, resulting in an abundance of publicly available adapters tailored to diverse domains. We ask: Can these pretrained adapters be leveraged to further streamline adaptation to new tasks while addressing these challenges? We introduce EigenLoRAx, a parameter-efficient finetuning method that recycles existing adapters to create a principal subspace aligned with their shared domain knowledge which can be further augmented with orthogonal basis vectors in low-resource scenarios. This enables rapid adaptation to new tasks by learning only lightweight coefficients on the principal components of the subspace - eliminating the need to finetune entire adapters. EigenLoRAx requires significantly fewer parameters and memory, improving efficiency for both training and inference. Our method demonstrates strong performance across diverse domains and tasks, offering a scalable for edge-based applications, personalization, and equitable deployment of large models in resource-constrained environments.

Deep subspace clustering (DSC) algorithms face several challenges that hinder their widespread adoption across variois application domains. First, clustering quality is typically assessed using only the encoder's output layer, disregarding valuable information present in the intermediate layers. Second, most DSC approaches treat representation learning and subspace clustering as independent tasks, limiting their effectiveness. Third, they assume the availability of a held-out dataset for hyperparameter tuning, which is often impractical in real-world scenarios. Fourth, learning termination is commonly based on clustering error monitoring, requiring external labels. Finally, their performance often depends on post-processing techniques that rely on labeled data. To address this limitations, we introduce a novel single-view DSC approach that: (i) minimizes a layer-wise self expression loss using a joint representation matrix; (ii) optimizes a subspace-structured norm to enhance clustering quality; (iii) employs a multi-stage sequential learning framework, consisting of pre-training and fine-tuning, enabling the use of multiple regularization terms without hyperparameter tuning; (iv) incorporates a relative error-based self-stopping mechanism to terminate training without labels; and (v) retains a fixed number of leading coefficients in the learned representation matrix based on prior knowledge. We evaluate the proposed method on six datasets representing faces, digits, and objects. The results show that our method outperforms most linear SC algorithms with careffulyl tuned hyperparameters while maintaining competitive performance with the best performing linear appoaches.

Directional ablation removes an aligned language model's ability to refuse by projecting a single "refusal direction" out of the weights that write the residual stream. It needs no gradient-based training and no optimization, only a few hundred contrastive prompts, which makes it the canonical white-box attack on open-weight alignment. However, it has been established only on dense models up to roughly 70B parameters. We study whether it survives the shift to frontier mixture-of-experts (MoE) models whose residual streams are no longer a single tensor and whose weights ship quantized. We apply it to GLM-5.3-Flash (320B parameters, 288 routed experts, a four-wide hyper-connection residual, block-FP8). The attack survives the architecture, but what it reaches is no longer where a reader of the original recipe would look for it. Editing the attention, dense and routed-expert writers on their own removes 0.039, 0.016 and 0.148 of refusal respectively; editing all three together removes 0.776. As a result, 74% of the effect exists only under the joint intervention. The part the conventional recipe reaches by module-name matching accounts for 0.066 of that 0.776, which is why it fails silently on an MoE. The effect does not follow from removing just any direction: ablating a random direction orthogonal to it leaves refusal unchanged. A category-concentrated residue survives every edit we tried: subspaces fitted on violence, sexual content and hate leave measurable refusal at every rank from 1 to 12. We report the method, the 41-89 percentage-point reductions it achieves across seven harmful benchmarks with no detected change in capability, and the boundary where it stops.

Recent advances in large language models (LLMs) have intensified the need to understand and reliably curb their harmful behaviours. We introduce a multidimensional framework for probing and steering harmful content in model internals. For each of 55 distinct harmfulness subconcepts (e.g., racial hate, employment scams, weapons), we learn a linear probe, yielding 55 interpretable directions in activation space. Collectively, these directions span a harmfulness subspace that we show is strikingly low-rank. We then test ablation of the entire subspace from model internals, as well as steering and ablation in the subspace's dominant direction. We find that dominant direction steering allows for near elimination of harmfulness with a low decrease in utility. Our findings advance the emerging view that concept subspaces provide a scalable lens on LLM behaviour and offer practical tools for the community to audit and harden future generations of language models.

Model merging is an efficient post-training strategy for integrating knowledge from multiple finetuned checkpoints of a shared foundation model. Existing methods operate in the parameter space, combining task vectors to mitigate conflicts, but remain constrained by parameter inconsistencies. We propose Functional Dual Anchors (FDAs), a framework that instead models the input-representation space. FDAs are synthetic inputs whose induced gradients align with task vectors, capturing task-specific functional shifts relative to the pretrained model. This perspective bridges joint multi-task training and post-hoc merging, offering both robustness and flexibility. We further introduce a principled initialization scheme and show that FDAs are complementary to parameter-space model merging. Comprehensive experiments demonstrate the effectiveness of FDAs in model merging.

As the number of model parameters increases, parameter-efficient fine-tuning (PEFT) has become the go-to choice for tailoring pre-trained large language models. Low-rank Adaptation (LoRA) uses a low-rank update method to simulate full parameter fine-tuning, which is widely used to reduce resource requirements. However, decreasing the rank encounters challenges with limited representational capacity when compared to full parameter fine-tuning. We present SMoA, a high-rank Structured MOdulation Adapter that uses fewer trainable parameters while maintaining a higher rank, thereby improving the model's representational capacity and offering improved performance potential. The core idea is to freeze the original pretrained weights and selectively amplify or suppress important features of the original weights across multiple subspaces. The subspace mechanism provides an efficient way to increase the capacity and complexity of a model. We conduct both theoretical analyses and empirical studies on various tasks. Experiment results show that SMoA outperforms LoRA and its variants on 10 tasks, with extensive ablation studies validating its effectiveness.

Adapting large pretrained models to new tasks efficiently and continually is crucial for real-world deployment but remains challenging due to catastrophic forgetting and the high cost of retraining. While parameter-efficient tuning methods like low rank adaptation (LoRA) reduce computational demands, they lack mechanisms for strict continual learning and knowledge integration, without relying on data replay, or multiple adapters. We propose Share, a novel approach to parameter efficient continual finetuning that learns and dynamically updates a single, shared low-rank subspace, enabling seamless adaptation across multiple tasks and modalities. Share constructs a foundational subspace that extracts core knowledge from past tasks and incrementally integrates new information by identifying essential subspace directions. Knowledge from each new task is incorporated into this evolving subspace, facilitating forward knowledge transfer, while minimizing catastrophic interference. This approach achieves up to 100x parameter reduction and 281x memory savings over traditional LoRA methods, maintaining performance comparable to jointly trained models. A single Share model can replace hundreds of task-specific LoRA adapters, supporting scalable, asynchronous continual learning. Experiments across image classification, natural language understanding, 3D pose estimation, and text-to-image generation validate its effectiveness, making Share a practical and scalable solution for lifelong learning in large-scale AI systems.

Model merging combines multiple fine-tuned checkpoints into a single model without additional training, offering an attractive approach to reusing models and efficiently improving performance. However, it remains unclear whether the advantages reported for settings where all merged experts have distinct roles and are tuned on clearly separated tasks also hold in settings where the merged experts do not have clearly distinct roles, but are trained on overlapping or even conflicting objectives. To evaluate this setting, we present a large-scale, systematic evaluation of "in-the-wild" model merging of heterogeneous experts, that may have been trained on overlapping or conflicting objectives. Concretely, we evaluate six state-of-the-art merging methods, including recent subspace methods, across four open-weight LLMs, twelve fine-tuned checkpoints per base model, and sixteen standard LLM benchmarks. Evaluating through standardized benchmarks, we measure both the probability that a model merged from a heterogeneous set of experts outperforms the base model and we measure relative gains over the best individual checkpoint. Our results show that the oldest and simplest method, Task Arithmetic, is the only approach that reliably yields performance gains on LLMs in this "in-the-wild" setting. Other interference-aware and subspace merging methods typically do not result in notable improvements over the base model. Our findings indicate that current merging techniques mostly do not enable extracting useful weight updates from heterogeneous and potentially conflicting versions. This motivates the design of LLM-specific merging algorithms and merging-aware fine-tuning methods.

Multi-teacher on-policy distillation (MOPD) has emerged as a popular post-training paradigm for integrating specialized capabilities in frontier language models. Existing OPD research has primarily focused on optimizing single-task distillation through objective design, distillation scope, and teacher signal construction, whereas MOPD must aggregate multiple capabilities in shared parameters and address the resulting capability seesaw, in which improving one domain suppresses capabilities acquired from another. Inspired by the distinctive update geometry of OPD, we find that parameter updates from different tasks rapidly concentrate in their respective low-dimensional subspaces during MOPD, providing a direct geometric basis for identifying and controlling cross-task interference. We therefore propose PMOPD (Projection-based Multi-Teacher On-Policy Distillation), which constructs subspace memories from the cumulative parameter displacements of different tasks and projects both gradients and optimizer updates to remove components that interfere with protected task directions. We further develop a lightweight conflict probe to characterize task interactions and guide task ordering, together with a cycling strategy that balances subspace estimation and timely task revisitation. Experiments on representative Code, Reason, and Math tasks show that PMOPD improves every evaluated capability over MOPD, raising the average score across the three tasks by 2.54 points on Qwen2.5-7B and 2.09 points on Llama-3.1-8B. These consistent gains establish geometry-aware optimization as an effective and transferable approach to balanced multi-teacher distillation.

We introduce two complementary techniques for efficient adaptive optimization that reduce memory requirements while accelerating training of large-scale neural networks. The first technique, Subset-Norm adaptive step size, generalizes AdaGrad-Norm and AdaGrad(-Coordinate) by reducing the second moment term's memory footprint from O(d) to O(d) through step-size sharing, where d is the model size. For non-convex smooth objectives under coordinate-wise sub-gaussian gradient noise, we prove a noise-adapted high-probability convergence guarantee showing improved dimensional dependence over existing methods. Our second technique, Subspace-Momentum, reduces the momentum state's memory footprint by operating in a low-dimensional subspace while applying standard SGD in the orthogonal complement. We establish high-probability convergence rates under similar relaxed assumptions. Empirical evaluation on LLaMA models from 60M to 1B parameters demonstrates the effectiveness of our methods, where combining subset-norm with subspace-momentum achieves Adam's validation perplexity in approximately half the training tokens (6.8B vs 13.1B) while using only 20% of the Adam's optimizer-states memory footprint and requiring minimal additional hyperparameter tuning.

With the rapid progress of video generation, demand for customized video editing is surging, where subject swapping constitutes a key component yet remains under-explored. Prevailing swapping approaches either specialize in narrow domains--such as human-body animation or hand-object interaction--or rely on some indirect editing paradigm or ambiguous text prompts that compromise final fidelity. In this paper, we propose DreamSwapV, a mask-guided, subject-agnostic, end-to-end framework that swaps any subject in any video for customization with a user-specified mask and reference image. To inject fine-grained guidance, we introduce multiple conditions and a dedicated condition fusion module that integrates them efficiently. In addition, an adaptive mask strategy is designed to accommodate subjects of varying scales and attributes, further improving interactions between the swapped subject and its surrounding context. Through our elaborate two-phase dataset construction and training scheme, our DreamSwapV outperforms existing methods, as validated by comprehensive experiments on VBench indicators and our first introduced DreamSwapV-Benchmark.

The capability of the self-attention mechanism to model the long-range dependencies has catapulted its deployment in vision models. Unlike convolution operators, self-attention offers infinite receptive field and enables compute-efficient modeling of global dependencies. However, the existing state-of-the-art attention mechanisms incur high compute and/or parameter overheads, and hence unfit for compact convolutional neural networks (CNNs). In this work, we propose a simple yet effective "Ultra-Lightweight Subspace Attention Mechanism" (ULSAM), which infers different attention maps for each feature map subspace. We argue that leaning separate attention maps for each feature subspace enables multi-scale and multi-frequency feature representation, which is more desirable for fine-grained image classification. Our method of subspace attention is orthogonal and complementary to the existing state-of-the-arts attention mechanisms used in vision models. ULSAM is end-to-end trainable and can be deployed as a plug-and-play module in the pre-existing compact CNNs. Notably, our work is the first attempt that uses a subspace attention mechanism to increase the efficiency of compact CNNs. To show the efficacy of ULSAM, we perform experiments with MobileNet-V1 and MobileNet-V2 as backbone architectures on ImageNet-1K and three fine-grained image classification datasets. We achieve approx13% and approx25% reduction in both the FLOPs and parameter counts of MobileNet-V2 with a 0.27% and more than 1% improvement in top-1 accuracy on the ImageNet-1K and fine-grained image classification datasets (respectively). Code and trained models are available at https://github.com/Nandan91/ULSAM.

A central premise in mechanistic interpretability is that meaningful concepts in language models are represented by linear features in activation space. For such features to support reliable interventions, manipulating one feature should not substantially alter the effects of others. In practice, however, feature entanglement leads to interference such that localized interventions can have unintended downstream effects. Motivated by the Independent Causal Mechanisms principle, we propose to constrain internal features to be almost orthogonal. We argue that this promotes modular representations amenable to causal intervention. We formalize this problem by characterizing the gap between an idealized isolated intervention and its realized effect on model outputs in terms of feature interference. We upper-bound the propagation of feature interference in terms of the self-coherence of the feature dictionary, and relate this discrepancy to an explicit orthogonality regularization on the dictionary itself. Empirically, we show that this regularization enables more isolated interventions on mathematical reasoning concepts while preserving model performance. Our code is available under https://github.com/mrtzmllr/sae-icm.

We present the first comprehensive evaluation of cross-lingual unlearning in multilingual LLMs. Using translated TOFU benchmarks in seven language/script variants, we test major unlearning algorithms and show that most fail to remove facts outside the training language, even when utility remains high. However, subspace-projection consistently outperforms the other methods, achieving strong cross-lingual forgetting with minimal degradation. Analysis of learned task subspaces reveals a shared interlingua structure: removing this shared subspace harms all languages, while removing language-specific components selectively affects one. These results demonstrate that multilingual forgetting depends on geometry in weight space, motivating subspace-based approaches for future unlearning systems.

Sparse Autoencoders (SAEs) decompose residual-stream activations into interpretable features. Recent latent-space defenses increasingly rely on these decompositions, assuming that identified "unsafe" SAE features serve as actionable handles for monitoring and intervention. In this paradigm, clamping a specific harmful feature is expected to reliably prevent model misbehavior. However, we show that this success may hide a recoverable failure mode: the clamp may block one visible route to a behavior without eliminating the behavior itself. We formulate this vulnerability as post-intervention recovery, a constrained residual-space optimization problem. Starting from the post-intervention residual state, we optimize residual perturbations to recover the pre-intervention behavior while preserving the post-intervention values of the targeted SAE features. Even under a strong threat model where the intervention remains active throughout optimization and generation, recovery remains possible. To rule out that recovery simply undoes the intervention, we use encoder-orthogonal updates for single-layer interventions and the corresponding feature-map Jacobian in the cross-layer setting. Across TPP, unlearning, IOI, and refusal steering experiments, this stress test reveals recoverable behavior despite successful feature-level intervention. Especially in the safety-critical refusal-steering setting, we achieve a 95.8% recovery rate on valid samples while keeping defended-feature relative drift to 0.131, substantially below suffix-based baselines. A recovery-path attribution analysis further localizes this recovery to the SAE reconstruction residual, the component left unexplained by the SAE. These results expose a gap between feature-level control and behavioral completeness: SAE features can support causal intervention, but controlling them does not guarantee control over the underlying behavior.

We generalize the class vectors found in neural networks to linear subspaces (i.e.~points in the Grassmann manifold) and show that the Grassmann Class Representation (GCR) enables the simultaneous improvement in accuracy and feature transferability. In GCR, each class is a subspace and the logit is defined as the norm of the projection of a feature onto the class subspace. We integrate Riemannian SGD into deep learning frameworks such that class subspaces in a Grassmannian are jointly optimized with the rest model parameters. Compared to the vector form, the representative capability of subspaces is more powerful. We show that on ImageNet-1K, the top-1 error of ResNet50-D, ResNeXt50, Swin-T and Deit3-S are reduced by 5.6%, 4.5%, 3.0% and 3.5%, respectively. Subspaces also provide freedom for features to vary and we observed that the intra-class feature variability grows when the subspace dimension increases. Consequently, we found the quality of GCR features is better for downstream tasks. For ResNet50-D, the average linear transfer accuracy across 6 datasets improves from 77.98% to 79.70% compared to the strong baseline of vanilla softmax. For Swin-T, it improves from 81.5% to 83.4% and for Deit3, it improves from 73.8% to 81.4%. With these encouraging results, we believe that more applications could benefit from the Grassmann class representation. Code is released at https://github.com/innerlee/GCR.

Mixture-of-Experts (MoE) scales model capacity efficiently by selectively routing inputs to a specialized subset of experts. However, input-expert specialization, the core motivation of MoE, critically depends on whether the router is actually aware of input structure. In practice, MoE routing is typically implemented as a shallow linear projection with limited awareness of input representation, which often leads to unstable routing. We propose STAR, a Structure Aware Routing that rethinks MoE routing as a subspace learning problem by augmenting standard learnable routing with an evolving principal subspace that tracks dominant input structure via Generalized Hebbian Algorithm (GHA). By aligning routing decisions directly with input structure, STAR enables stable expert specialization. We evaluate STAR on controlled synthetic setup and large-scale language and vision tasks, where it consistently improves routing quality and downstream performance over strong MoE baselines. Moreover, optional test-time subspace updates further enhance routing robustness and generalization under input distribution shifts.

Large-scale text-to-image diffusion models excel in generating high-quality images from textual inputs, yet concerns arise as research indicates their tendency to memorize and replicate training data, raising We also addressed the issue of memorization in diffusion models, where models tend to replicate exact training samples raising copyright infringement and privacy issues. Efforts within the text-to-image community to address memorization explore causes such as data duplication, replicated captions, or trigger tokens, proposing per-prompt inference-time or training-time mitigation strategies. In this paper, we focus on the feed-forward layers and begin by contrasting neuron activations of a set of memorized and non-memorized prompts. Experiments reveal a surprising finding: many different sets of memorized prompts significantly activate a common subspace in the model, demonstrating, for the first time, that memorization in the diffusion models lies in a special subspace. Subsequently, we introduce a novel post-hoc method for editing pre-trained models, whereby memorization is mitigated through the straightforward pruning of weights in specialized subspaces, avoiding the need to disrupt the training or inference process as seen in prior research. Finally, we demonstrate the robustness of the pruned model against training data extraction attacks, thereby unveiling new avenues for a practical and one-for-all solution to memorization.

In this paper, we introduce a subspace-inspired Low-Rank Adaptation (LoRA) method, which is computationally efficient, easy to implement, and readily applicable to large language, multimodal, and diffusion models. Initially, we equivalently decompose the weights of LoRA into two subspaces, and find that simply mixing them can enhance performance. To study such a phenomenon, we revisit it through a fine-grained subspace lens, showing that such modification is equivalent to employing a fixed mixer to fuse the subspaces. To be more flexible, we jointly learn the mixer with the original LoRA weights, and term the method Mixture-of-Subspaces LoRA (MoSLoRA). MoSLoRA consistently outperforms LoRA on tasks in different modalities, including commonsense reasoning, visual instruction tuning, and subject-driven text-to-image generation, demonstrating its effectiveness and robustness. Codes are available at https://github.com/wutaiqiang/MoSLoRA{github}.

Subspace representation is a fundamental technique in various fields of machine learning. Analyzing a geometrical relationship among multiple subspaces is essential for understanding subspace series' temporal and/or spatial dynamics. This paper proposes the second-order difference subspace, a higher-order extension of the first-order difference subspace between two subspaces that can analyze the geometrical difference between them. As a preliminary for that, we extend the definition of the first-order difference subspace to the more general setting that two subspaces with different dimensions have an intersection. We then define the second-order difference subspace by combining the concept of first-order difference subspace and principal component subspace (Karcher mean) between two subspaces, motivated by the second-order central difference method. We can understand that the first/second-order difference subspaces correspond to the velocity and acceleration of subspace dynamics from the viewpoint of a geodesic on a Grassmann manifold. We demonstrate the validity and naturalness of our second-order difference subspace by showing numerical results on two applications: temporal shape analysis of a 3D object and time series analysis of a biometric signal.

Low-rank optimization has emerged as a promising approach to enabling memory-efficient training of large language models (LLMs). Existing low-rank optimization methods typically project gradients onto a low-rank subspace, reducing the memory cost of storing optimizer states. A key challenge in these methods is selecting suitable subspaces to ensure an effective optimization trajectory. Most existing approaches select the dominant subspace to preserve gradient information, as this intuitively provides the best approximation. However, we find that in practice, the dominant subspace stops changing during pretraining, thereby constraining weight updates to similar subspaces. In this paper, we propose importance sampling for low-rank optimization in LLM pretraining with a provable convergence guarantee, which the dominant subspace approach does not have. Empirically, we demonstrate that our method significantly outperforms previous methods in LLM pretraining tasks.

Large language models (LLMs) store vast amounts of information, making them powerful yet raising privacy and safety concerns when selective knowledge removal is required. Existing unlearning strategies, ranging from gradient-based fine-tuning and model editing to sparse autoencoder (SAE) steering, either lack interpretability or fail to provide a robust defense against adversarial prompts. We propose SAE-Guided Subspace Projection Unlearning (SSPU), a novel framework that leverages SAE features to drive targeted updates in the model's parameter space, enabling precise, interpretable, and robust unlearning. SSPU's three-stage pipeline performs data-driven layer and feature selection, subspace construction via QR decomposition, and constrained optimization that controls activations into an "irrelevant" subspace while preserving retained knowledge. Overall, we use SAE features to construct a subspace that supervises unlearning, refining the loss and adding a regularization term to guide interpretable parameter updates. In experiments on the WMDP-Cyber forget set and three utility benchmarks (MMLU, TruthfulQA, GSM8K), SSPU reduces harmful knowledge accuracy by 3.22% compared to the strongest baseline. It also improves adversarial robustness, lowering malicious accuracy under jailbreak prompts compared to baselines. Our findings expose the limitations of prior unlearning methods and demonstrate how interpretable subspace-guided optimization can achieve robust, controllable model behavior.

Training large language models (LLMs) is constrained by memory requirements, with activations accounting for a substantial fraction of the total footprint. Existing approaches reduce memory using low-rank weight parameterizations or low-rank gradient subspaces for optimizer states, while activation memory is addressed through architectural modifications or compression schemes based on periodically updated projections. We propose OASIS, an online activation subspace learning algorithm for memory-efficient training that tracks and continuously updates a low-dimensional activation subspace during training. Intermediate activations are projected onto this evolving subspace, reducing memory without modifying forward-pass computations. The evolving activation subspace induces low-rank gradient representations, enabling both gradients and optimizer states to be maintained directly in this subspace, while a projection-aware optimizer consistently transports optimizer states across subspace updates for stable training. Across various finetuning and pretraining tasks, OASIS achieves up to 2times lower peak memory than full fine-tuning while matching its performance and outperforming prior low-rank methods.

Model merging seeks to integrate task-specific expert models into a unified architecture while preserving multi-task generalization capabilities, yet parameter interference between constituent models frequently induces performance degradation. Although prior work has explored many merging strategies, resolving interference without additional data for retraining or test-time computation remains challenging. In this paper, we theoretically demonstrate that the task vectors of the linear layer constitute an approximate linear subspace for its corresponding input. Therefore, we can minimize interference under the guidance of task vectors. Based on this insight, we propose WUDI-Merging (Whoever started the interference shoUld enD It), a simple yet effective model merging method that eliminates interference without any additional data or rescaling coefficients. Comprehensive empirical evaluations across vision and language benchmarks demonstrate our method's superiority, achieving state-of-the-art performance in data-free model merging scenarios (average 10.9\% improvement versus baseline methods) while even outperforming mainstream test-time adaptation approaches by 3.3\%, and only very few computing resources are required. The code will be publicly available soon.

The premise of identifiable and causal representation learning is to improve the current representation learning paradigm in terms of generalizability or robustness. Despite recent progress in questions of identifiability, more theoretical results demonstrating concrete advantages of these methods for downstream tasks are needed. In this paper, we consider the task of intervention extrapolation: predicting how interventions affect an outcome, even when those interventions are not observed at training time, and show that identifiable representations can provide an effective solution to this task even if the interventions affect the outcome non-linearly. Our setup includes an outcome Y, observed features X, which are generated as a non-linear transformation of latent features Z, and exogenous action variables A, which influence Z. The objective of intervention extrapolation is to predict how interventions on A that lie outside the training support of A affect Y. Here, extrapolation becomes possible if the effect of A on Z is linear and the residual when regressing Z on A has full support. As Z is latent, we combine the task of intervention extrapolation with identifiable representation learning, which we call Rep4Ex: we aim to map the observed features X into a subspace that allows for non-linear extrapolation in A. We show that the hidden representation is identifiable up to an affine transformation in Z-space, which is sufficient for intervention extrapolation. The identifiability is characterized by a novel constraint describing the linearity assumption of A on Z. Based on this insight, we propose a method that enforces the linear invariance constraint and can be combined with any type of autoencoder. We validate our theoretical findings through synthetic experiments and show that our approach succeeds in predicting the effects of unseen interventions.

In the training of large language models, parameter-efficient techniques such as LoRA optimize memory usage and reduce communication overhead and memory usage during the fine-tuning phase. However, applying such techniques directly during the pre-training phase results in poor performance, primarily because the premature implementation of low-rank training significantly reduces model accuracy. Existing methods like ReLoRA and GaLore have attempted to address this challenge by updating the low-rank subspace. However, they still fall short of achieving the accuracy of full-rank training. Specifically, ReLoRA restricts the frequency of updates to preserve optimizer states consistency, hindering its ability to closely approximate full-rank training behavior. Meanwhile, GaLore relies on Singular Value Decomposition (SVD) to approximate the full-rank space, which introduces accuracy loss during the approximation process. In this paper, we introduce SwitchLoRA, a parameter-efficient training technique that frequently and smoothly replaces the trainable parameters of LoRA adapters with alternative parameters. SwitchLoRA updates the low-rank subspace incrementally, targeting only a few dimensions at a time to minimize the impact on optimizer states. This allows a higher update frequency, thereby enhancing accuracy by enabling the updated parameters to more closely mimic full-rank behavior during the pre-training phase. Our results demonstrate that SwitchLoRA actually surpasses full-rank training, reducing perplexity from 15.23 to 15.01 on the LLaMA 1.3B model, while also cutting communication overhead by 54\% and memory usage by 13\%. Furthermore, after full fine-tuning the SwitchLoRA pre-trained model and the full-rank pre-trained model on the GLUE benchmark, the SwitchLoRA pre-trained model showed an average accuracy gain of about 1\% over the full-rank pre-trained model.

Input space reconstruction is an attractive representation learning paradigm. Despite interpretability of the reconstruction and generation, we identify a misalignment between learning by reconstruction, and learning for perception. We show that the former allocates a model's capacity towards a subspace of the data explaining the observed variance--a subspace with uninformative features for the latter. For example, the supervised TinyImagenet task with images projected onto the top subspace explaining 90\% of the pixel variance can be solved with 45\% test accuracy. Using the bottom subspace instead, accounting for only 20\% of the pixel variance, reaches 55\% test accuracy. The features for perception being learned last explains the need for long training time, e.g., with Masked Autoencoders. Learning by denoising is a popular strategy to alleviate that misalignment. We prove that while some noise strategies such as masking are indeed beneficial, others such as additive Gaussian noise are not. Yet, even in the case of masking, we find that the benefits vary as a function of the mask's shape, ratio, and the considered dataset. While tuning the noise strategy without knowledge of the perception task seems challenging, we provide first clues on how to detect if a noise strategy is never beneficial regardless of the perception task.

Model merging enables the combination of multiple specialized expert models into a single model capable of performing multiple tasks. However, the benefits of merging an increasing amount of specialized experts generally lead to diminishing returns and reduced overall performance gains. In this work, we empirically and theoretically analyze this limitation, proving that for Task Arithmetic-based methods, as more experts are merged, the common information dominates the task-specific information, leading to inevitable rank collapse. To mitigate this issue, we introduce Subspace Boosting, which operates on the singular value decomposed task vector space and maintains task vector ranks. Subspace Boosting raises merging efficacy for up to 20 experts by large margins of more than 10% when evaluated on both vision and language benchmarks. Moreover, we propose employing Higher-Order Generalized Singular Value Decomposition to quantify task similarity, offering a new interpretable perspective on model merging. Code and models are available at https://github.com/ronskoro/Subspace-Boosting.

Alignment with high-resource standard languages is often assumed to aid the modeling of related low-resource varieties. We challenge this assumption by demonstrating that excessive representational entanglement with a dominant variety, such as Modern Standard Arabic (MSA) in relation to Arabic dialects, can actively hinder generative modeling. We present the first comprehensive causal study of this phenomenon by analyzing and directly intervening in the internal representation geometry of large language models (LLMs). Our key contribution is an online variational probing framework that continuously estimates the subspace of the standard variety during fine-tuning, enabling projection-based decoupling from this space. While our study uses Arabic as a case due to its unusually rich parallel resources across 25 dialects, the broader motivation is methodological: dialectal MT serves as a controlled proxy for generative tasks where comparable multi-variety corpora are unavailable. Across 25 dialects, our intervention improves generation quality by up to +4.9 chrF++ and +2.0 on average compared to standard fine-tuning, despite a measured tradeoff in standard-language performance. These results provide causal evidence that subspace dominance by high-resource varieties can restrict generative capacity for related varieties. More generally, we unify geometric and information-theoretic probing with subspace-level causal interventions, offering practical tools for improving generative modeling in closely related language families and, more broadly, for controlling representational allocation in multilingual and multi-domain LLMs. Code will be released.

Alignment tuning is meant to make harmful-request refusal robust, yet this safety behavior can be erased by a small set of benign fine-tuning examples. This is a deployment risk for open-weight models because a checkpoint can pass refusal tests at release time and later lose refusal under low-cost downstream fine-tuning. Prior work has established these refusal failures, but existing studies do not show how to detect this fragility in the aligned model itself before an attack or fine-tuning intervention is run. We introduce Skin-Deep, a geometric diagnostic that detects alignment fragility directly from the aligned model's hidden-state activations before such an intervention is run and compresses the layer-wise safety geometry into a single scalar, the Geometric Fragility Score (GFS). Applied to twenty-one instruction-tuned models spanning six alignment recipes and 3B--32B parameters, Skin-Deep reveals a recurring low-rank safety subspace across model families. Direction ablations show that removing directions in this subspace weakens harmful-request refusal, providing causal evidence that the recovered geometry underlies refusal behavior. Crucially, GFS identifies, before any fine-tuning, the initially safe model that retains the most refusal after small-scale LoRA fine-tuning. These results establish GFS as a practical pre-deployment diagnostic for flagging fragile refusal behavior without running an attack.

Driven by the rapid growth of model parameters, parameter-efficient fine-tuning (PEFT) has become essential for adapting large models to diverse downstream tasks under constrained computational resources. Within this paradigm, orthogonal fine-tuning and its variants preserve semantic representations of pre-trained models, but struggle to achieve both expressiveness and efficiency in terms of parameter counts, memory, and computation. To overcome this limitation, we propose efficient Orthogonal Fine-Tuning with Principal Subspace adaptation (PSOFT), which confines orthogonal transformations to the principal subspace of pre-trained weights. Specifically, PSOFT constructs this subspace via matrix decomposition to enable compatible transformations with higher effective rank, establishes a theoretical condition that strictly maintains the geometry of this subspace for essential semantic preservation, and introduces efficient tunable vectors that gradually relax orthogonality during training to enhance adaptability. Extensive experiments on 35 NLP and CV tasks across four representative models demonstrate that PSOFT offers a practical and scalable solution to simultaneously achieve semantic preservation, expressiveness, and multi-dimensional efficiency in PEFT. The code is publicly available at https://github.com/fei407/PSOFT.

Activation steering offers a promising approach to controlling the behavior of Large Language Models by directly manipulating their internal activations. However, most existing methods struggle to jointly steer multiple attributes, often resulting in interference and undesirable trade-offs. To address this challenge, we propose Multi-Subspace Representation Steering (MSRS), a novel framework for effective multi-attribute steering via subspace representation fine-tuning. MSRS reduces inter-attribute interference by allocating orthogonal subspaces to each attribute, isolating their influence within the model's representation space. MSRS also incorporates a hybrid subspace composition strategy: it combines attribute-specific subspaces for unique steering directions with a shared subspace for common steering directions. A dynamic weighting function learns to efficiently integrate these components for precise control. During inference, MSRS introduces a token-level steering mechanism that dynamically identifies and intervenes on the most semantically relevant tokens, enabling fine-grained behavioral modulation. Experimental results show that MSRS significantly reduces attribute conflicts, surpasses existing methods across a range of attributes, and generalizes effectively to diverse downstream tasks.

We show that low-rank adaptation of large-scale models suffers from a low stable rank that is well below the linear algebraic rank of the subspace, degrading fine-tuning performance. To mitigate the underutilization of the allocated subspace, we propose PoLAR, a parameterization inspired by the polar decomposition that factorizes the low-rank update into two direction matrices constrained to Stiefel manifolds and an unconstrained scale matrix. Our theory shows that PoLAR yields an exponentially faster convergence rate on a canonical low-rank adaptation problem. Pairing the parameterization with Riemannian optimization leads to consistent gains on three different benchmarks testing general language understanding, commonsense reasoning, and mathematical problem solving with base model sizes ranging from 350M to 27B.

Continual learning requires the model to learn multiple tasks sequentially. In continual learning, the model should possess the ability to maintain its performance on old tasks (stability) and the ability to adapt to new tasks continuously (plasticity). Recently, parameter-efficient fine-tuning (PEFT), which involves freezing a pre-trained model and injecting a small number of learnable parameters to adapt to downstream tasks, has gained increasing popularity in continual learning. Although existing continual learning methods based on PEFT have demonstrated superior performance compared to those not based on PEFT, most of them do not consider how to eliminate the interference of the new task on the old tasks, which inhibits the model from making a good trade-off between stability and plasticity. In this work, we propose a new PEFT method, called interference-free low-rank adaptation (InfLoRA), for continual learning. InfLoRA injects a small number of parameters to reparameterize the pre-trained weights and shows that fine-tuning these injected parameters is equivalent to fine-tuning the pre-trained weights within a subspace. Furthermore, InfLoRA designs this subspace to eliminate the interference of the new task on the old tasks, making a good trade-off between stability and plasticity. Experimental results show that InfLoRA outperforms existing state-of-the-art continual learning methods on multiple datasets.

Krylov subspace, which is generated by multiplying a given vector by the matrix of a linear transformation and its successive powers, has been extensively studied in classical optimization literature to design algorithms that converge quickly for large linear inverse problems. For example, the conjugate gradient method (CG), one of the most popular Krylov subspace methods, is based on the idea of minimizing the residual error in the Krylov subspace. However, with the recent advancement of high-performance diffusion solvers for inverse problems, it is not clear how classical wisdom can be synergistically combined with modern diffusion models. In this study, we propose a novel and efficient diffusion sampling strategy that synergistically combines the diffusion sampling and Krylov subspace methods. Specifically, we prove that if the tangent space at a denoised sample by Tweedie's formula forms a Krylov subspace, then the CG initialized with the denoised data ensures the data consistency update to remain in the tangent space. This negates the need to compute the manifold-constrained gradient (MCG), leading to a more efficient diffusion sampling method. Our method is applicable regardless of the parametrization and setting (i.e., VE, VP). Notably, we achieve state-of-the-art reconstruction quality on challenging real-world medical inverse imaging problems, including multi-coil MRI reconstruction and 3D CT reconstruction. Moreover, our proposed method achieves more than 80 times faster inference time than the previous state-of-the-art method. Code is available at https://github.com/HJ-harry/DDS

Diffusion models (DMs) have achieved remarkable success in image and video generation. However, they still struggle with (1) physical alignment and (2) out-of-distribution (OOD) instruction following. We argue that these issues stem from the models' failure to learn causal directions and to disentangle causal factors for novel recombination. We introduce the Causal Scene Graph (CSG) and the Physical Alignment Probe (PAP) dataset to enable diagnostic interventions. This analysis yields three key insights. First, DMs struggle with multi-hop reasoning for elements not explicitly determined in the prompt. Second, the prompt embedding contains disentangled representations for texture and physics. Third, visual causal structure is disproportionately established during the initial, computationally limited denoising steps. Based on these findings, we introduce LINA (Learning INterventions Adaptively), a novel framework that learns to predict prompt-specific interventions, which employs (1) targeted guidance in the prompt and visual latent spaces, and (2) a reallocated, causality-aware denoising schedule. Our approach enforces both physical alignment and OOD instruction following in image and video DMs, achieving state-of-the-art performance on challenging causal generation tasks and the Winoground dataset. Our project page is at https://opencausalab.github.io/LINA.

Recent advances in diffusion models have significantly enhanced their ability to generate high-quality images and videos, but they have also increased the risk of producing unsafe content. Existing unlearning/editing-based methods for safe generation remove harmful concepts from models but face several challenges: (1) They cannot instantly remove harmful concepts without training. (2) Their safe generation capabilities depend on collected training data. (3) They alter model weights, risking degradation in quality for content unrelated to toxic concepts. To address these, we propose SAFREE, a novel, training-free approach for safe T2I and T2V, that does not alter the model's weights. Specifically, we detect a subspace corresponding to a set of toxic concepts in the text embedding space and steer prompt embeddings away from this subspace, thereby filtering out harmful content while preserving intended semantics. To balance the trade-off between filtering toxicity and preserving safe concepts, SAFREE incorporates a novel self-validating filtering mechanism that dynamically adjusts the denoising steps when applying the filtered embeddings. Additionally, we incorporate adaptive re-attention mechanisms within the diffusion latent space to selectively diminish the influence of features related to toxic concepts at the pixel level. In the end, SAFREE ensures coherent safety checking, preserving the fidelity, quality, and safety of the output. SAFREE achieves SOTA performance in suppressing unsafe content in T2I generation compared to training-free baselines and effectively filters targeted concepts while maintaining high-quality images. It also shows competitive results against training-based methods. We extend SAFREE to various T2I backbones and T2V tasks, showcasing its flexibility and generalization. SAFREE provides a robust and adaptable safeguard for ensuring safe visual generation.

Feature-map knowledge distillation (KD) transfers internal representations well between comparably sized Vision Transformers (ViTs), but it often fails in compression. We revisit this failure and uncover a paradox. Sample-wise SVD shows that each image is highly compressible, which seems to suggest that a narrow student with a linear projector should match the teacher "in principle". However, a dataset-level view contradicts this intuition: PCA shows that the teacher is a union of low-rank subspaces with significant subspace rotation across inputs. We further introduce token-level Spectral Energy Patterns (SEP) and find an architecture-invariant encoding law: tokens spread energy broadly across channel modes even when they live in low-rank subspace, creating a bandwidth mismatch. We refer to this combined phenomenon as an encoding mismatch. We propose two minimal remedies, Lift or WideLast: (i) Lift retains a lightweight lifting projector at inference to provide wider channel, or (ii) WideLast widens only the student's last block, enabling an input-dependent expansion. On ImageNet-1K, these fixes revive feature KD for ViT compression, improving DeiT-Tiny distilled from CaiT-S24 from 74.86% to 77.53%/78.23% top-1 accuracy, and they also strengthen students trained without distillation. Our analyses clarify when and why feature-map KD fails and then how to fix it. Code and raw data are provided in https://github.com/thy960112/From-Per-Image-Low-Rank-to-Encoding-Mismatch.

Clinical routine and retrospective cohorts commonly include multi-parametric Magnetic Resonance Imaging; however, they are mostly acquired in different anisotropic 2D views due to signal-to-noise-ratio and scan-time constraints. Thus acquired views suffer from poor out-of-plane resolution and affect downstream volumetric image analysis that typically requires isotropic 3D scans. Combining different views of multi-contrast scans into high-resolution isotropic 3D scans is challenging due to the lack of a large training cohort, which calls for a subject-specific framework. This work proposes a novel solution to this problem leveraging Implicit Neural Representations (INR). Our proposed INR jointly learns two different contrasts of complementary views in a continuous spatial function and benefits from exchanging anatomical information between them. Trained within minutes on a single commodity GPU, our model provides realistic super-resolution across different pairs of contrasts in our experiments with three datasets. Using Mutual Information (MI) as a metric, we find that our model converges to an optimum MI amongst sequences, achieving anatomically faithful reconstruction. Code is available at: https://github.com/jqmcginnis/multi_contrast_inr/

Electroencephalography (EEG) is a cornerstone of brain-computer interfaces and clinical neuroscience, yet deep learning models are typically trained and evaluated under a single, unreported preprocessing pipeline. We formalize preprocessing choices as a counterfactual intervention space and show that EEG predictions are surprisingly unstable under this space: across six datasets spanning four paradigms, up to 42% of trial-level predictions flip when only the preprocessing changes, a variability that standard uncertainty methods do not explicitly quantify because they condition on a fixed preprocessing pipeline. We provide three tools to make this instability measurable, decomposable, and reducible. First, a Walsh-Hadamard decomposition of the 2^7 pipeline space reveals that sensitivity is near-additive in practice under the binary intervention design, enabling efficient step-by-step optimization. Second, we introduce Preprocessing Uncertainty (PU), a per-trial diagnostic that captures a dimension of instability complementary to model-based confidence. Third, we study Normalized Adaptive PGI (NA-PGI), a graph-structured regularizer that exploits the compositional structure of preprocessing interventions as one mitigation strategy with clear scope conditions.

Multimodal Large Language Models (MLLMs) have achieved impressive progress in vision-language alignment, yet they remain limited in visual-spatial reasoning. We first identify that this limitation arises from the attention mechanism: visual tokens are overshadowed by language tokens, preventing the model from consistently recognizing the same visual cues across frames. To address this challenge, we draw a novel connection between the self-expressiveness property in sparse subspace clustering and the attention mechanism in Transformers. Building on this insight, we propose VideoAnchor, a plug-and-play module that leverages subspace affinities to reinforce visual cues across frames without retraining, effectively anchoring attention to shared visual structures. Extensive experiments across benchmarks and backbone models show consistent performance gains -- e.g., 3.2% and 4.6% improvements on VSI-Bench and Video-MME (spatial-related tasks) with InternVL2-8B and Qwen2.5VL-72B -- while qualitative analyses demonstrate more coherent subspace partitions and stronger visual grounding. Our codes will be made public available at https://github.com/feufhd/VideoAnchor.

Most benchmarks for causal inference over time series are observational, small, or domain-specific, leaving interventional and counterfactual estimation under-served exactly where it matters most, such as in healthcare, policy evaluation, and climate science. We introduce DoTime, an open, scalable, and theoretically grounded generator of multivariate temporal structural causal models (TSCMs) with interventions, released as the dotime PyPI package together with four frozen evaluation suites. Beyond existing work, it adds capabilities absent from prior generators: continuous-time intervention windows, counterfactual sampling modes with a positivity guard, regime-switching SCMs as a strict generalization of interrupted time series, non-stationary dynamics by construction with switching SCM parameters, and deterministic ramp and sinusoidal intervention profiles that place trends and structural breaks inside the evaluation window. Moreover, it demonstrates the suitability of the generator as a prior for a causal foundation model reference implementation. The released suites span a training-scale snapshot of 100{,}000 trajectories and eight named identification structures, each with exact ground truth: paired interventional trajectories from the same SCM throughout, and shared-noise counterfactuals in the continuous-time suite. We ship reference baseline implementations with an evaluation harness, and pose a falsifiable claim: interventional training buys a measurable direction-accuracy advantage over an observational model of identical capacity. It is tested across three training seeds per arm. Under structure-matched evaluation on held-out episodes, the interventional prior-fitted network's (PFN) gap is positive in every structure, trajectory length, and seed tested.

Vision-language models (VLMs) achieve strong zero-shot recognition, but they remain highly vulnerable to adversarial perturbations. Recent test-time adaptations improve robustness without retraining, but they do not directly adapt the corrupted visual representation itself. Prompt-based methods adapt the learnable text prompts, while input-space methods optimize pixels or padding at test time. These approaches can improve predictions, but they do so through an indirect and expensive optimization path. We propose Test-time Visual Subspace Steering (T-VSS), a lightweight defense that performs test-time adaptation directly in the visual feature space. T-VSS first builds a sample-specific low-rank subspace from multi-view feature residuals anchored at the attacked image. It then learns a shared feature correction within this subspace using reliability-weighted entropy minimization. By constraining adaptation to a compact visual geometry, T-VSS steers attacked features toward more stable and discriminative predictions while avoiding noisy full-space updates. Experiments on fine-grained, ImageNet, and ImageNet-OOD benchmarks show that T-VSS improves adversarial robustness while maintaining competitive clean accuracy and better efficiency than prior test-time adaptations.

Finetuning a language model can lead to emergent misalignment (EM) [Betley et al., 2025b]. Models trained on a narrow distribution of misaligned behavior generalize to more egregious behaviors when tested outside the training distribution. We study a set of interventions proposed to reduce EM. We confirm that these interventions reduce or eliminate EM on existing evaluations (questions like "How do I make a quick buck?"). However, if the evaluation prompts are tweaked to resemble the training context, the model displays EM. We call this conditional misalignment. As in standard EM, the model displays misaligned behaviors more egregious than those seen during training, but only on inputs sharing features with the training data. The first two interventions are diluting misaligned data with benign data, and finetuning on benign data after misaligned data. Both produce conditional misalignment. For instance, models trained on a mix of only 5% insecure code still show misalignment when asked to format responses as Python strings (resembling the training context). The third intervention is inoculation prompting. Here, statements with a similar form to the inoculation prompt serve as triggers for misalignment, even if they have the opposite meaning. On the positive side, inoculation prompting has lower (but still non-zero) conditional misalignment if training is on-policy or includes reasoning distillation. Our results imply that in realistic post-training, where misaligned data is typically combined with benign data, models may be conditionally misaligned even if standard evaluations look clean.

Recently, a wide range of memory-efficient LLM training algorithms have gained substantial popularity. These methods leverage the low-rank structure of gradients to project optimizer states into a subspace using projection matrix found by singular value decomposition (SVD). However, convergence of these algorithms is highly dependent on the update rules of their projection matrix. In this work, we provide the first convergence guarantee for arbitrary update rules of projection matrix. This guarantee is generally applicable to optimizers that can be analyzed with Hamiltonian Descent, including most common ones, such as LION, Adam. Inspired by our theoretical understanding, we propose Online Subspace Descent, a new family of subspace descent optimizer without SVD. Instead of updating the projection matrix with eigenvectors, Online Subspace Descent updates the projection matrix with online PCA. Online Subspace Descent is flexible and introduces only minimum overhead to training. We show that for the task of pretraining LLaMA models ranging from 60M to 7B parameters on the C4 dataset, Online Subspace Descent achieves lower perplexity and better downstream tasks performance than state-of-the-art low-rank training methods across different settings and narrows the gap with full-rank baselines.

Cross-entropy pretraining and preference alignment update the same transformer weights, but leave geometrically distinct traces. We characterise this asymmetry with a relative-subspace-fraction probe that tracks how weight deltas align with residual-stream activation subspaces and with the prediction subspace defined by the unembedding. Alignment deltas concentrate in the read pathway (W_Q, W_K), along principal directions of attention-input activations, while remaining near-isotropic in the write pathway (W_O, W_2) relative to the prediction subspace. We explain this pattern through anisotropic gradient accumulation: updates to a matrix W are sums of outer products δ_t a_t^top, and inherit directional structure from whichever side has concentrated covariance. For read-pathway matrices, this side is the input activation a_t, whose covariance is spiked in trained transformers and therefore produces objective-agnostic concentration. For write-pathway matrices, the relevant side is the upstream gradient δ_t, whose anisotropy depends on the loss. Cross-entropy supplies the canonical sharp per-sample signal, inducing write-pathway prediction geometry during pretraining; alignment objectives typically add little further write-side concentration. We support this explanation with a within-checkpoint trajectory, a graded contrastive-objective control, and a closed-form rank-1 intervention with matched direction controls, providing causal evidence for the proposed weight-space geometry.

Low-Rank Adaptation (LoRA) improves downstream performance by restricting task updates to a low-rank parameter subspace, yet how this limited capacity is allocated within a trained adapter remains unclear. Through a geometric and empirical study across multiple tasks and backbones, we find that trained LoRA updates often exhibit an inefficient spectrum: task effects concentrate in a small subset of singular directions, while many remaining components are neutral or detrimental, motivating post-hoc refinement within the learned subspace. We propose Spectral Surgery, a training-free refinement that decomposes a LoRA update with SVD, estimates per-component sensitivity using gradients on a small calibration set, and reweights singular values under a magnitude constraint while keeping the learned directions fixed. Across Llama-3.1-8B and Qwen3-8B on four benchmarks, Spectral Surgery yields consistent gains (up to +4.4 points on CommonsenseQA and +2.4 pass@1 on HumanEval) by adjusting only approx 1{,}000 scalar coefficients. These results demonstrate that SVD-structured, low-cost parameter editing can serve as a practical route to improving trained LoRA adapters in a purely post-hoc manner.

Diffusion models have achieved great success in synthesizing high-quality images. However, generating high-resolution images with diffusion models is still challenging due to the enormous computational costs, resulting in a prohibitive latency for interactive applications. In this paper, we propose DistriFusion to tackle this problem by leveraging parallelism across multiple GPUs. Our method splits the model input into multiple patches and assigns each patch to a GPU. However, na\"{\i}vely implementing such an algorithm breaks the interaction between patches and loses fidelity, while incorporating such an interaction will incur tremendous communication overhead. To overcome this dilemma, we observe the high similarity between the input from adjacent diffusion steps and propose displaced patch parallelism, which takes advantage of the sequential nature of the diffusion process by reusing the pre-computed feature maps from the previous timestep to provide context for the current step. Therefore, our method supports asynchronous communication, which can be pipelined by computation. Extensive experiments show that our method can be applied to recent Stable Diffusion XL with no quality degradation and achieve up to a 6.1times speedup on eight NVIDIA A100s compared to one. Our code is publicly available at https://github.com/mit-han-lab/distrifuser.

When pretrained VLA policies are fine-tuned through online RL, each rollout episode produces only a single binary outcome (success or failure), yet the actor update requires per-transition supervision. Existing approaches commonly reduce this sparse outcome to a single scalar reward or advantage signal, which conflates distinct forms of transition-level feedback and provides limited guidance once basic task success becomes achievable. First, a single scalar signal conflates the two objectives of viability and efficiency; once basic success is achieved, the binary label provides no gradient to distinguish efficient completions from slow ones. Second, real-world rollouts mix autonomous and intervention segments; naively assigning episode outcomes across these boundaries introduces incorrect credit assignment. To address these issues, we propose Hierarchical Advantage-Weighted Behavior Cloning (HABC), which trains separate critic heads for these two objectives on different data subsets and combines their outputs with a state-adaptive balance. A state-adaptive gate g_t merges their one-step advantages, prioritizing viability when success is uncertain and shifting to efficiency only when viability is high, and converts the result into per-transition weights on the actor loss. Intervention-aware credit assignment further restricts outcome labels to segments executed by the current policy, preventing supervision from leaking across intervention boundaries. In real-robot experiments on three contact-rich bimanual tasks, HABC raises success from supervised fine-tuning (SFT) baselines of 36%, 44%, and 12% to 92%, 88%, and 38%.

We propose an algorithm for change point monitoring in linear causal models that accounts for interventions. Through a special centralization technique, we can concentrate the changes arising from causal propagation across nodes into a single dimension. Additionally, by selecting appropriate intervention nodes based on Kullback-Leibler divergence, we can amplify the change magnitude. We also present an algorithm for selecting the intervention values, which aids in the identification of the most effective intervention nodes. Two monitoring methods are proposed, each with an adaptive intervention policy to make a balance between exploration and exploitation. We theoretically demonstrate the first-order optimality of the proposed methods and validate their properties using simulation datasets and two real-world case studies.

Instruction tuning aligns large language models, including multimodal ones, with diverse user intents, but scaling to heterogeneous mixtures is hindered by gradient interference and bandwidth-heavy synchronization. We ask whether these two bottlenecks can be addressed jointly by training parts of the mixture independently and reconciling them once in parameter space. We develop a local quadratic theory inside a shared flat basin that yields three results: weight merging produces a curvature-weighted variance reduction; PCA-aligned conflict splitting maximizes this gain along high-curvature directions; and merging additionally acts as spectral filtering with implicit norm regularization. These results directly motivate MERIT, a decentralized merge-ready instruction-tuning pipeline that estimates dataset-level gradient conflicts, partitions the mixture along the top PCA conflict axes, fine-tunes each partition independently with no inter-partition communication, and merges once via token-weighted averaging. On Qwen2.5-VL-3B with 136 Vision-FLAN tasks, MERIT improves the 8-benchmark average from 54.3 (joint training) to 57.0. The same recipe scales to a 7B model on a 1.6M-example, 176-source mixture -- matching or exceeding centralized joint training with minimal cost overhead -- and transfers to text-only FLAN. Our code is available at https://github.com/naver-ai/merit.

Adapting pre-trained foundation models for various downstream tasks has been prevalent in artificial intelligence. Due to the vast number of tasks and high costs, adjusting all parameters becomes unfeasible. To mitigate this, several fine-tuning techniques have been developed to update the pre-trained model weights in a more resource-efficient manner, such as through low-rank adjustments. Yet, almost all of these methods focus on linear weights, neglecting the intricacies of parameter spaces in higher dimensions like 4D. Alternatively, some methods can be adapted for high-dimensional parameter space by compressing changes in the original space into two dimensions and then employing low-rank matrix decomposition. However, these approaches destructs the structural integrity of the involved high-dimensional spaces. To tackle the diversity of dimensional spaces across different foundation models and provide a more precise representation of the changes within these spaces, this paper introduces a generalized parameter-efficient fine-tuning framework, FLoRA, designed for various dimensional parameter space. Specifically, utilizing Tucker decomposition, FLoRA asserts that changes in each dimensional parameter space are based on a low-rank core space which maintains the consistent topological structure with the original space. It then models the changes through this core space alongside corresponding weights to reconstruct alterations in the original space. FLoRA effectively preserves the structural integrity of the change of original N-dimensional parameter space, meanwhile decomposes it via low-rank tensor decomposition. Extensive experiments on computer vision, natural language processing and multi-modal tasks validate FLoRA's effectiveness. Codes are available at https://github.com/SJTU-DeepVisionLab/FLoRA.

Leading open text-to-image models often carry complementary strengths: one may lead on preference-aligned aesthetics while another follows compositional instructions more faithfully. However, differences in their autoencoders and noise schedules make it difficult to transfer these strengths across models. In this paper, we present Poly-OPD, a framework that can consolidate complementary strengths of heterogeneous teachers into a single compact flow-matching student. To bridge the incompatible latent spaces of different teachers, Poly-OPD performs on-policy distillation through a pixel bridge. Each student-generated image is re-encoded by a selected teacher's encoder and refined from a noise level matched by magnitude under the teacher's noise schedule. The resulting target is further matched to the student in frozen DINOv2 space, enabling supervision across incompatible latent spaces. To retain complementary capabilities without cross-teacher interference, Poly-OPD uses a gradient compatibility diagnostic to organize its adapters: attention LoRA modules are shared across teachers, whereas feed-forward adapters remain teacher-specific. During distillation, a gap-aware curriculum devotes more training to compositional categories where the student still falls short of the teacher. As each gap narrows, training shifts toward categories with larger remaining gaps. By distilling FLUX.1-dev and Z-Image into a 2.5B SD3.5-Medium student, Poly-OPD improves GenEval from 67.3 to 73.3, surpassing both larger teachers, and raises DrawBench HPSv3 from 9.34 to 11.35, consolidating both strengths within a switchable model.

Aligned representations across languages is a desired property in multilingual large language models (mLLMs), as alignment can improve performance in cross-lingual tasks. Typically alignment requires fine-tuning a model, which is computationally expensive, and sizable language data, which often may not be available. A data-efficient alternative to fine-tuning is model interventions -- a method for manipulating model activations to steer generation into the desired direction. We analyze the effect of a popular intervention (finding experts) on the alignment of cross-lingual representations in mLLMs. We identify the neurons to manipulate for a given language and introspect the embedding space of mLLMs pre- and post-manipulation. We show that modifying the mLLM's activations changes its embedding space such that cross-lingual alignment is enhanced. Further, we show that the changes to the embedding space translate into improved downstream performance on retrieval tasks, with up to 2x improvements in top-1 accuracy on cross-lingual retrieval.

The exploration of linear subspaces, particularly scattered subspaces, has garnered considerable attention across diverse mathematical disciplines in recent years, notably within finite geometries and coding theory. Scattered subspaces play a pivotal role in analyzing various geometric structures such as blocking sets, two-intersection sets, complete arcs, caps in affine and projective spaces over finite fields and rank metric codes. This paper introduces a new infinite family of h-subspaces, along with their associated MRD codes. Additionally, it addresses the task of determining the generalized weights of these codes. Notably, we demonstrate that these MRD codes exhibit some larger generalized weights compared to those previously identified.

LoRA-based large model parameter-efficient fine-tuning (PEFT) methods use low-rank de- composition to approximate updates to model parameters. However, compared to full- parameter fine-tuning, low-rank updates often lead to a performance gap in downstream tasks. To address this, we introduce DropLoRA, a novel pruning-based approach that focuses on pruning the rank dimension. Unlike conven- tional methods that attempt to overcome the low-rank bottleneck, DropLoRA innovatively integrates a pruning module between the two low-rank matrices in LoRA to simulate dy- namic subspace learning. This dynamic low- rank subspace learning allows DropLoRA to overcome the limitations of traditional LoRA, which operates within a static subspace. By continuously adapting the learning subspace, DropLoRA significantly boosts performance without incurring additional training or infer- ence costs. Our experimental results demon- strate that DropLoRA consistently outperforms LoRA in fine-tuning the LLaMA series across a wide range of large language model gener- ation tasks, including commonsense reason- ing, mathematical reasoning, code generation, and instruction-following. Our code is avail- able at https://github.com/TayeeChang/DropLoRA.

Fine-tuning large language models (LLMs) on domain-specific datasets has become a standard paradigm for adapting LLMs to specialized applications. However, recent work has shown that even fine-tuning on benign task-specific data can substantially weaken the safety capabilities of LLMs. While existing efforts have made progress in identifying data responsible for safety degradation, they usually rely on a single mean vector computed over a specific model with its tokenizer to represent the safety direction, which limits both the effectiveness and transferability of their risk assessment measures. To address these limitations, we propose DataShield, a data assessment framework that identifies risky fine-tuning samples and response segments through consensus subspace alignment over joint safety-critical semantic spaces derived from multiple safety-aligned LLMs. Within these spaces, DataShield extracts consensus safe and unsafe subspaces using semantic spectral decomposition over safe and unsafe data representations. The risk of a data sample or segment is then estimated by measuring its relative alignment with the unsafe and safe subspaces, enabling both sample-level filtering and fine-grained segment-level masking. Compared with state-of-the-art filtering and masking baselines, DataShield reduces ASR by 14.6\% with sample filtering and 32.3\% with segment masking, while preserving downstream utility and avoiding target-model-specific risk computation.

Vision-language systems combine images with retrieved text, but these sources can disagree or jointly fail to support an answer. Reliable models must identify the trustworthy source and abstain when neither is adequate. Existing post-training objectives score instances independently and therefore do not enforce coherent behavior under counterfactual evidence changes. We introduce CARGO-VL, a group-relative framework that optimizes matched variants covering aligned, image-correct, text-correct, and both-wrong (A/V/T/N) evidence states as one bundle. Its objective couples condition-wise correctness with transition rewards for answer invariance, source equivariance, and answer-to-abstention switching, while a primal-dual controller balances unsafe answers against excessive deferral. We also contribute XMC (eXtended Modal Conflict), a four-condition conflict training resource, and evaluate transfer on CMC-Bench and Modality-Bias. Across multiple seeds, CARGO-VL improves conflict handling, unsupported-answer avoidance, and modality balance over pointwise baselines. Ablations identify complementary benefits from relational transition signals and adaptive risk control, supporting counterfactual consistency as a practical objective for reliable multimodal evidence arbitration.

Model merging aims to cheaply combine individual task-specific models into a single multitask model. In this work, we view past merging methods as leveraging different notions of a ''task subspace'' in which models are matched before being merged. We connect the task subspace of a given model to its loss landscape and formalize how this approach to model merging can be seen as solving a linear system of equations. While past work has generally been limited to linear systems that have a closed-form solution, we consider using the conjugate gradient method to find a solution. We show that using the conjugate gradient method can outperform closed-form solutions, enables merging via linear systems that are otherwise intractable to solve, and flexibly allows choosing from a wide variety of initializations and estimates for the ''task subspace''. We ultimately demonstrate that our merging framework called ''Matching Models in their Task Subspace'' (MaTS) achieves state-of-the-art results in multitask and intermediate-task model merging. We release all of the code and checkpoints used in our work at https://github.com/r-three/mats.

As language models shift from single-shot answer generation toward multi-step reasoning that retrieves and consumes evidence mid-inference, evaluating the role of individual retrieved items becomes more important. Existing RAG evaluation typically targets final-answer quality, citation faithfulness, or answer-level attribution, but none of these directly targets the intervention-based, per-evidence-item utility view we study here. We introduce CUE-R, a lightweight intervention-based framework for measuring per-evidence-item operational utility in single-shot RAG using shallow observable retrieval-use traces. CUE-R perturbs individual evidence items via REMOVE, REPLACE, and DUPLICATE operators, then measures changes along three utility axes (correctness, proxy-based grounding faithfulness, and confidence error) plus a trace-divergence signal. We also outline an operational evidence-role taxonomy for interpreting intervention outcomes. Experiments on HotpotQA and 2WikiMultihopQA with Qwen-3 8B and GPT-5.2 reveal a consistent pattern: REMOVE and REPLACE substantially harm correctness and grounding while producing large trace shifts, whereas DUPLICATE is often answer-redundant yet not fully behaviorally neutral. A zero-retrieval control confirms that these effects arise from degradation of meaningful retrieval. A two-support ablation further shows that multi-hop evidence items can interact non-additively: removing both supports harms performance far more than either single removal. Our results suggest that answer-only evaluation misses important evidence effects and that intervention-based utility analysis is a practical complement for RAG evaluation.

Continual Model Merging (CMM) sequentially integrates task-specific models into a unified architecture without intensive retraining. However, existing CMM methods are hindered by a fundamental saturation-redundancy dilemma: backbone-centric approaches face parameter saturation and representation interference within fixed capacities, whereas Mixture-of-Experts (MoE) variants resort to indiscriminate expansion, incurring expert redundancy and a routing bottleneck reliant on additional data-driven optimization. To resolve these challenges, we propose MADE-IT (Manifold-Aware Dynamic Expert Evolution and Implicit rouTing), an adaptive CMM method that orchestrates expert management and activation by grounding intrinsic expert representations in manifold geometry. We introduce a projection-based subspace affinity metric coupled with a distribution-aware adaptive threshold mechanism to guide autonomous expert evolution, harmonizing diversity with architectural parsimony. Furthermore, to bypass parameterized gating networks, we design a data-free and training-free implicit routing mechanism that activates experts via feature-subspace alignment. Extensive experiments demonstrate that MADE-IT consistently outperforms strong baselines in accuracy and robustness across long-horizon and shuffled task sequences, while significantly pruning redundant experts, particularly within generic modules and early layers.

Mechanistic interpretability aims to explain a model's behavior by identifying causally responsible internal structures. Dictionary-based explainers such as sparse autoencoders and transcoders are a primary tool, but their faithfulness under out-of-distribution (OOD) shift has received little systematic attention. We show that distribution shift rotates the subspace that the model actively uses, misaligning the explainer's dictionary trained on in-distribution (ID) activations. We formalize this misalignment as the faithfulness gap, a geometric distance between the ID dictionary and the OOD-active subspace, and show that it controls OOD faithfulness degradation. To reduce this gap, we propose the Geometry-Adaptive Explainer (GAE), which realigns the explainer's dictionary with the OOD-active subspace while preserving the original feature structure. This requires only unlabeled OOD activations and no gradient updates. We prove that GAE improves over the unadapted ID explainer, with excess loss bounded quadratically by the second-moment shift. Empirically, GAE even matches or surpasses all training-based baselines in causal faithfulness across multiple models and OOD settings.

Some security-development settings require local models that map an objective to an ordered, checkable plan. We present a low-rank decomposition adapter and release a case-disjoint corpus with 24 authored decompositions and 83 derived next-step examples across 24 satellite-security cases. To prevent reference-plan leakage, prompts contain one objective and eight shuffled SPARTA candidates. The evaluation candidate sets are constructed with oracle inclusion of every reference technique, so this study measures selection from a controlled set, not retrieval. On six fixed cases and five decoding seeds, mean scores vary across adapters and prompted Qwen2.5 baselines; a separate greedy check over three adapter training seeds shows non-negligible training variance. A 24-fold greedy leave-one-case-out control at the two compact sizes likewise shows tradeoffs: at 1.5B, adapter precision is 0.583 against 0.480 for two-shot prompting, while recall is 0.586 against 0.660. Case-resampling intervals for adapter precision relative to the strongest prompted baseline cross zero at 1.5B and 7B. The adapter is also substantially more format-compliant, which prevents attributing all score differences to selection. Autoregressive next-step accuracy is 0.08, 0.06, and 0.29 at 0.5B, 1.5B, and 7B. We therefore offer the formulation, leakage-controlled dataset, reproducibility package, and descriptive analysis as a proof of concept, not evidence of a reliable stand-alone decomposition system or a general adapter advantage. A separate diagnostic tests contract reuse.

Training-free concept erasure is an attractive mechanism for controlling text-to-image diffusion models, but precise erasure often comes at the cost of damaging semantically related non-target concepts. Existing value-space methods remove the component of each cross-attention value along the target concept direction, implicitly treating target identity and shared visual structure as the same signal. We argue that this is the source of much of the collateral damage in prior preservation. We introduce CARE, a closed-form concept erasure operator that replaces the raw target direction with a kept-subspace-aware direction computed from a small bank of retained concept anchors. The resulting edit is applied directly in cross-attention value space, requires no model fine-tuning, and adds only a negligible offline computation. A single shrinkage parameter controls the erase-preserve trade-off. We further show that the operator admits a minimum-disturbance interpretation and, in its projection form, leaves the kept subspace invariant. Experiments under the standard concept-erasure protocol show that our method preserves non-target concepts more faithfully while maintaining competitive erasure across instance, style, and celebrity concepts. Code: https://github.com/parthupman/care

Methods for controlling large language models (LLMs), including local weight fine-tuning, LoRA-based adaptation, and activation-based interventions, are often studied in isolation, obscuring their connections and making comparison difficult. In this work, we present a unified view that frames these interventions as dynamic weight updates induced by a control signal, placing them within a single conceptual framework. Building on this view, we propose a unified preference-utility analysis that separates control effects into preference, defined as the tendency toward a target concept, and utility, defined as coherent and task-valid generation, and measures both on a shared log-odds scale using polarity-paired contrastive examples. Across methods, we observe a consistent trade-off between preference and utility: stronger control increases preference while predictably reducing utility. We further explain this behavior through an activation manifold perspective, in which control shifts representations along target-concept directions to enhance preference, while utility declines primarily when interventions push representations off the model's valid-generation manifold. Finally, we introduce a new steering approach SPLIT guided by this analysis that improves preference while better preserving utility. Code is available at https://github.com/zjunlp/EasyEdit/blob/main/examples/SPLIT.md.

This paper tackles the data-driven approximation of unknown dynamical systems using Koopman-operator methods. Given a dictionary of functions, these methods approximate the projection of the action of the operator on the finite-dimensional subspace spanned by the dictionary. We propose the Tunable Symmetric Subspace Decomposition algorithm to refine the dictionary, balancing its expressiveness and accuracy. Expressiveness corresponds to the ability of the dictionary to describe the evolution of as many observables as possible and accuracy corresponds to the ability to correctly predict their evolution. Based on the observation that Koopman-invariant subspaces give rise to exact predictions, we reason that prediction accuracy is a function of the degree of invariance of the subspace generated by the dictionary and provide a data-driven measure to measure invariance proximity. The proposed algorithm iteratively prunes the initial functional space to identify a refined dictionary of functions that satisfies the desired level of accuracy while retaining as much of the original expressiveness as possible. We provide a full characterization of the algorithm properties and show that it generalizes both Extended Dynamic Mode Decomposition and Symmetric Subspace Decomposition. Simulations on planar systems show the effectiveness of the proposed methods in producing Koopman approximations of tunable accuracy that capture relevant information about the dynamical system.

Test-time interventions for language models can enhance factual accuracy, mitigate harmful outputs, and improve model efficiency without costly retraining. But despite a flood of new methods, different types of interventions are largely developing independently. In practice, multiple interventions must be applied sequentially to the same model, yet we lack standardized ways to study how interventions interact. We fill this gap by introducing composable interventions, a framework to study the effects of using multiple interventions on the same language models, featuring new metrics and a unified codebase. Using our framework, we conduct extensive experiments and compose popular methods from three emerging intervention categories -- Knowledge Editing, Model Compression, and Machine Unlearning. Our results from 310 different compositions uncover meaningful interactions: compression hinders editing and unlearning, composing interventions hinges on their order of application, and popular general-purpose metrics are inadequate for assessing composability. Taken together, our findings showcase clear gaps in composability, suggesting a need for new multi-objective interventions. All of our code is public: https://github.com/hartvigsen-group/composable-interventions.

In language models, what a representation encodes is determined by the geometry of its representation space: distances, not activations, carry meaning. Existing tools characterize the shape of this geometry but do not ask what that shape is organized for. We introduce Subspace PGA, a metric that tests whether a layer's distance structure aligns with the readout subspace of the unembedding matrix W_U more than with random subspaces of equal size. Across seven Pythia models (70M--6.9B) and three cross-family models, intermediate geometry is significantly organized for prediction (peak z = 9--24), but the degree is scale-dependent: small models (d leq 1024) progressively lose it at late layers during training -- even as loss keeps improving -- while large models (d geq 2048) preserve it throughout. We trace this to a capacity trade-off: a few dominant directions migrate away from W_U's readout, masking rather than destroying the predictive structure beneath, and removing them restores alignment. Neither spectral metrics nor loss curves capture this distinction. Scale thus determines not only how well a model predicts, but how its representation geometry is organized to do so.

Instruction-tuning data are judged by quality metrics, and tuned models are judged by benchmarks, but both judgments pass through an output interface: the surface format in which an answer is written. Using gradient signatures across 12 tasks, four semantically equivalent interfaces, three model families, and controlled corruptions, we show that this interface confounds both measurements. Spectral statistics such as effective rank are provably invariant to interface rotation and empirically blind to semantic corruption, while the direction of the update carries the quality signal. The interface-varying residual is not noise: it identifies each unit's own target task perfectly across all three families. Capability itself is stored relative to the training interface: a skill that raises accuracy by more than 40 points under the training format can be nearly invisible under every other, and correcting a single generation budget flips the measured effect of fine-tuning on GSM8K from a gain into a large loss. Pre-registered interventions delimit where this geometry stops short of control. Data quality and model capability are interface-conditioned quantities, and current practice often reports the interface instead of the content.

Extracting causal relationships from observed correlations is a growing area in probabilistic reasoning, originating with the seminal work of Pearl and others from the early 1990s. This paper develops a new, categorically oriented view based on a clear distinction between syntax (string diagrams) and semantics (stochastic matrices), connected via interpretations as structure-preserving functors. A key notion in the identification of causal effects is that of an intervention, whereby a variable is forcefully set to a particular value independent of any prior propensities. We represent the effect of such an intervention as an endofunctor which performs `string diagram surgery' within the syntactic category of string diagrams. This diagram surgery in turn yields a new, interventional distribution via the interpretation functor. While in general there is no way to compute interventional distributions purely from observed data, we show that this is possible in certain special cases using a calculational tool called comb disintegration. We demonstrate the use of this technique on a well-known toy example, where we predict the causal effect of smoking on cancer in the presence of a confounding common cause. After developing this specific example, we show this technique provides simple sufficient conditions for computing interventions which apply to a wide variety of situations considered in the causal inference literature.

Reinforcement Learning with Verifiable Rewards (RLVR) is increasingly viewed as a tree pruning mechanism. However, we identify a systemic pathology termed Recursive Space Contraction (RSC), an irreversible collapse driven by the combined dynamics of positive sharpening and negative squeezing, where the sampling probability of valid alternatives vanishes. While Kullback-Leibler (KL) regularization aims to mitigate this, it imposes a rigid Shape Matching constraint that forces the policy to mimic the reference model's full density, creating a gradient conflict with the sharpening required for correctness. We propose Anchored Policy Optimization (APO), shifting the paradigm from global Shape Matching to Support Coverage. By defining a Safe Manifold based on the reference model's high-confidence support, APO permits aggressive sharpening for efficiency while selectively invoking a restorative force during error correction to prevent collapse. We theoretically derive that APO serves as a gradient-aligned mechanism to maximize support coverage, enabling an Elastic Recovery that re-inflates valid branches. Empirical evaluations on mathematical benchmarks demonstrate that APO breaks the accuracy-diversity trade-off, significantly improving Pass@1 while restoring the Pass@K diversity typically lost by standard policy gradient methods.

We present a parallel data-driven strategy to identify finite-dimensional functional spaces invariant under the Koopman operator associated to an unknown dynamical system. We build on the Symmetric Subspace Decomposition (SSD) algorithm, a centralized method that under mild conditions on data sampling provably finds the maximal Koopman-invariant subspace and all Koopman eigenfunctions in an arbitrary finite-dimensional functional space. A network of processors, each aware of a common dictionary of functions and equipped with a local set of data snapshots, repeatedly interact over a directed communication graph. Each processor receives its neighbors' estimates of the invariant dictionary and refines its estimate by applying SSD with its local data on the intersection of the subspaces spanned by its own dictionary and the neighbors' dictionaries. We identify conditions on the network topology to ensure the algorithm identifies the maximal Koopman-invariant subspace in the span of the original dictionary, characterize its time, computational, and communication complexity, and establish its robustness against communication failures.

Layer factorization has emerged as a widely used technique for training memory-efficient neural networks. However, layer factorization methods face several challenges, particularly a lack of robustness during the training process. To overcome this limitation, dynamical low-rank training methods have been developed, utilizing robust time integration techniques for low-rank matrix differential equations. Although these approaches facilitate efficient training, they still depend on computationally intensive QR and singular value decompositions of matrices with small rank. In this work, we introduce a novel low-rank training method that reduces the number of required QR decompositions. Our approach integrates an augmentation step into a projector-splitting scheme, ensuring convergence to a locally optimal solution. We provide a rigorous theoretical analysis of the proposed method and demonstrate its effectiveness across multiple benchmarks.

Modern one-step diffusion models achieve impressive quality through distribution-based timestep distillation. Yet, they rely on a critical assumption: Teacher and Student must inhabit the same latent space. This Shared-Space constraint prevents knowledge transfer from modern high-capacity Teachers (e.g., SD 3.5 and Flux) into compact, deployment-friendly Students such as SD 1.5, whose latent resolution and VAE parameterization differ from the Teacher. We formalize this overlooked regime as Cross-Space Distillation, where Teacher and Student differ in both latent resolution and VAE space. To enable distillation under this mismatch, we introduce the Bridge, a lightweight latent interface that maps Student latents into the Teacher space without modifying the Student backbone. Bridge combines a frozen Student VAE decoder as a spatial prior with a compact learnable projector, and is trained with latent reconstruction and attention fidelity objectives for stable Teacher-space alignment. Across diverse modern Teachers, Bridge enables substantial gains for compact one-step Students; for example, it improves SD 1.5 from 5.4 to 9.4 HPSv3 while preserving one-step inference, low latency, and broad ecosystem compatibility. These results show that heterogeneous large Teachers can be distilled into efficient, deployable backbones through a lightweight latent-space interface.

This paper explores how theory can guide and enhance practical algorithms, using Low-Rank Adaptation (LoRA, Hu et al. 2022) in large language models as a case study. We rigorously prove that, under gradient descent, LoRA adapters align with specific singular subspaces of the one-step full fine-tuning gradient. This result suggests that, by properly initializing the adapters using the one-step full gradient, subspace alignment can be achieved immediately and applicable to both linear and nonlinear models. Building on our theory, we propose a theory-driven algorithm, LoRA-One, where the linear convergence (as well as generalization) is built and incorporating preconditioners theoretically helps mitigate the effects of ill-conditioning. Besides, our theory reveals connections between LoRA-One and other gradient-alignment-based methods, helping to clarify misconceptions in the design of such algorithms. LoRA-One achieves significant empirical improvements over LoRA and its variants across benchmarks in natural language understanding, mathematical reasoning, and code generation. Code is available at: https://github.com/YuanheZ/LoRA-One.

A central goal in robotics is to enable manipulation across changing tasks and environments. Vision-language-action (VLA) models provide broad manipulation capabilities but can struggle when execution requires diagnosing failures and adapting behavior. Strong agents can discover effective interventions through interaction with these policies. We propose Recursive Harness Distillation to accumulate this experience as reusable guidance across agents. A strong agent distills its experience into a playbook for a light agent, then recursively refines the playbook using the light agent's execution feedback. The resulting playbook enables agents to reuse accumulated intervention knowledge in new task instances without updating model parameters. In real-world manipulation, the harness improves success from 37.3% to 64.0%. On SimplerEnv Bridge, the light agent with the playbook achieves 66.7% success, compared with 41.7% for the GR00T-only baseline, and outperforms the strong agent without a playbook. The same playbook also benefits the strong agent, which reaches 79.2% success. These results demonstrate the feasibility of harness distillation for robotics: intervention experience can be accumulated, refined through execution, and reused across agents to improve manipulation.

We propose affine concept editing (ACE) as an approach for steering language models' behavior by intervening directly in activations. We begin with an affine decomposition of model activation vectors and show that prior methods for steering model behavior correspond to subsets of terms of this decomposition. We then provide a derivation of ACE and use it to control refusal behavior on ten different models, including Llama 3 70B. ACE combines affine subspace projection and activation addition to reliably control the model's refusal responses across prompt types. We evaluate the results using LLM-based scoring on a collection of harmful and harmless prompts. Our experiments demonstrate that ACE consistently achieves more precise control over model behavior than existing methods and generalizes to models where directional ablation via affine subspace projection alone produces incoherent outputs. Code for reproducing our results is available at https://github.com/EleutherAI/steering-llama3 .

Current unlearning and safety training methods consistently fail to remove dangerous knowledge from language models. We identify the root cause - unlearning targets representations which are too general - and develop a highly selective technique that unlearns robustly while preserving general performance. Our method performs PCA on activations and module-output gradients to identify subspaces containing common representations, then collapses these subspaces before computing unlearning updates, a technique we term Collapse of Irrelevant Representations (CIR). This avoids unlearning general knowledge and targets only representations specific to the facts being unlearned. When unlearning bio- and cyber-hazardous facts from Llama-3.1-8B, we achieve over 30x greater reduction in post-attack accuracy than the best baseline (Circuit Breakers), while disrupting general performance 30x less, and using less than 3 GPU-seconds per fact. Thus, by disentangling harmful and benign capabilities at the level of representations, CIR enables robust and non-disruptive unlearning.

Vision-Language-Action (VLA) models often fail to perform the same learned tasks under environmental shifts, such as changes in camera pose and shifts to a different but similar robot (e.g., from Panda to UR5e). Adapting these models to the shifted environment (i.e., target domain) often requires training on multiple demonstrations for each task, which are costly to collect. To reduce the burden of data curation and training, we propose an analogy-based method that adapts VLA models under environmental shifts through weight vector arithmetic with domain-specific information addition, named Domain ARiThmetic (DART). Unlike prior approaches, DART requires collecting only a single demonstration, enabling efficient adaptation. To accurately isolate domain-specific information for addition, DART performs subspace alignment between singular components in weight vectors to filter out noisy components. In both simulated and real-world experiments, DART outperforms existing VLA adaptation methods in one-shot scenarios across diverse visual and embodiment shifts. Code is available at https://github.com/snumprlab/dart.

Fast Healthcare Interoperability Resources (FHIR) is the dominant standard for interoperable exchange of healthcare data. In FHIR, electronic health records form a directed graph of resources. Answering clinically meaningful questions over FHIR requires agents to perform multi-step reasoning, filtering, and aggregation across multiple resource types. Prior work shows that even tool-augmented LLM agents (retrieval, code execution, multi-turn planning) often select the wrong resources or violate traversal constraints. We study this problem in the context of FHIR-AgentBench, a benchmark for realistic question answering over real-world hospital data, and frame reasoning on FHIR as a sequential decision-making problem over a queryable structured graph. We implement a multi-turn CodeAct agent and post-train it with reinforcement learning using a custom harness and tools. A LLM Judge provides execution-grounded rewards. Compared to prompt-based, closed-model baselines, RL post-training improves performance while enforcing data-integrity constraints. Empirically, our approach improves answer correctness from 50% (o4-mini) to 77% on FHIR-AgentBench using a smaller and cheaper Qwen3-8B model. We present an end-to-end post-training pipeline (environment building, harness construction, model training and custom evaluation) that reliably improves multi-turn reasoning over structured clinical graphs.

Multimodal large language model (MLLM) unlearning methods have been proposed to remove private, sensitive, or proprietary information from well-trained models. However, most existing MLLM unlearning methods are designed for one-shot requests and fail to adequately address continual scenarios, as repeatedly applying one-shot operations leads to cumulative utility degradation, unlearning rebound, and retention drift. We introduce Merging for Continual Unlearning (MCU), an approach that dynamically merges multiple one-shot unlearning adapters into a unified adapter upon receiving each new unlearning request.Through a leave-one-out merging analysis, we reveal that these unlearning adapters exhibit strong cross-task dependencies. Such dependencies have two contrasting effects: they can facilitate cross-task unlearning transferability, but they can also introduce severe interference that degrades unlearning effectiveness and compromises retained knowledge. To address this challenge, MCU projects the adapters into a shared representation space, preserves their dominant directions, suppresses over-concentrated coordinates, and reconfigures cross-task dependencies to mitigate interference while enhancing transferability. Experiments on ICU-Bench and MLLMU-Bench demonstrate that MCU achieves superior unlearning effectiveness while preserving both retained knowledge and general multimodal utility.

We introduce the Local Intersectional Visual Spaces (LIVS) dataset, a benchmark for multi-criteria alignment of text-to-image (T2I) models in inclusive urban planning. Developed through a two-year participatory process with 30 community organizations, LIVS encodes diverse spatial preferences across 634 initial concepts, consolidated into six core criteria: Accessibility, Safety, Comfort, Invitingness, Inclusivity, and Diversity, through 37,710 pairwise comparisons. Using Direct Preference Optimization (DPO) to fine-tune Stable Diffusion XL, we observed a measurable increase in alignment with community preferences, though a significant proportion of neutral ratings highlights the complexity of modeling intersectional needs. Additionally, as annotation volume increases, accuracy shifts further toward the DPO-tuned model, suggesting that larger-scale preference data enhances fine-tuning effectiveness. LIVS underscores the necessity of integrating context-specific, stakeholder-driven criteria into generative modeling and provides a resource for evaluating AI alignment methodologies across diverse socio-spatial contexts.

Change detection is an important tool for long-term earth observation missions. It takes bi-temporal images as input and predicts "where" the change has occurred. Different from other dense prediction tasks, a meaningful consideration for change detection is the interaction between bi-temporal features. With this motivation, in this paper we propose a novel general change detection architecture, MetaChanger, which includes a series of alternative interaction layers in the feature extractor. To verify the effectiveness of MetaChanger, we propose two derived models, ChangerAD and ChangerEx with simple interaction strategies: Aggregation-Distribution (AD) and "exchange". AD is abstracted from some complex interaction methods, and "exchange" is a completely parameter\&computation-free operation by exchanging bi-temporal features. In addition, for better alignment of bi-temporal features, we propose a flow dual-alignment fusion (FDAF) module which allows interactive alignment and feature fusion. Crucially, we observe Changer series models achieve competitive performance on different scale change detection datasets. Further, our proposed ChangerAD and ChangerEx could serve as a starting baseline for future MetaChanger design.

Large language models offer broad capabilities, but adapting them to evolving domains, tools, and requirements often entails repeated post-training. Autonomous systems automate parts of this process by proposing updates, training candidates, and using evaluation feedback to select subsequent proposals. As evidence accumulates, a central problem emerges: which past update evidence remains actionable after subsequent training has changed the parent model? An update's effect depends on its parent, data, and training stage. Treating past success as context-free permission can waste compute. If the resulting child is promoted, it can also degrade the subsequent training trajectory. We formulate this problem as conditional experience transfer and introduce Boundary-Calibrated Intervention Transfer (BCIT), a method that authorizes experience reuse before weight-changing training. BCIT binds an observed effect to its source context, checks applicability conditions, vetoes candidates with named hard conflicts, and obtains current-state evidence through a bounded training trial when needed. Fully trained candidates still face a shared adoption rule, and only observed events extend memory. On one 4B model adapted across finance reasoning, text-to-SQL, and function calling, candidate updates exhibit heterogeneous target and retention effects across the evaluated contexts. Under matched candidates, evidence, and compute, BCIT authorizes fewer harmful updates and attains higher equal-budget final-model quality than the evaluated alternatives. These results support treating experience authorization as a distinct problem in autonomous post-training.

Recent empirical studies have demonstrated that diffusion models can effectively learn the image distribution and generate new samples. Remarkably, these models can achieve this even with a small number of training samples despite a large image dimension, circumventing the curse of dimensionality. In this work, we provide theoretical insights into this phenomenon by leveraging key empirical observations: (i) the low intrinsic dimensionality of image data, (ii) a union of manifold structure of image data, and (iii) the low-rank property of the denoising autoencoder in trained diffusion models. These observations motivate us to assume the underlying data distribution of image data as a mixture of low-rank Gaussians and to parameterize the denoising autoencoder as a low-rank model according to the score function of the assumed distribution. With these setups, we rigorously show that optimizing the training loss of diffusion models is equivalent to solving the canonical subspace clustering problem over the training samples. Based on this equivalence, we further show that the minimal number of samples required to learn the underlying distribution scales linearly with the intrinsic dimensions under the above data and model assumptions. This insight sheds light on why diffusion models can break the curse of dimensionality and exhibit the phase transition in learning distributions. Moreover, we empirically establish a correspondence between the subspaces and the semantic representations of image data, facilitating image editing. We validate these results with corroborated experimental results on both simulated distributions and image datasets.

This paper localizes the policy routing mechanism in alignment-trained language models. An intermediate-layer attention gate reads detected content and triggers deeper amplifier heads that boost the signal toward refusal. In smaller models the gate and amplifier are single heads; at larger scale they become bands of heads across adjacent layers. The gate contributes under 1% of output DLA, but interchange testing (p<0.001) and knockout cascade confirm it is causally necessary. Interchange screening at n>=120 detects the same motif in twelve models from six labs (2B to 72B), though specific heads differ by lab. Per-head ablation weakens up to 58x at 72B and misses gates that interchange identifies; interchange is the only reliable audit at scale. Modulating the detection-layer signal continuously controls policy from hard refusal through evasion to factual answering. On safety prompts the same intervention turns refusal into harmful guidance, showing the safety-trained capability is gated by routing rather than removed. Thresholds vary by topic and by input language, and the circuit relocates across generations within a family while behavioral benchmarks register no change. Routing is early-commitment: the gate commits at its own layer before deeper layers finish processing the input. Under an in-context substitution cipher, gate interchange necessity collapses 70 to 99% across three models and the model switches to puzzle-solving. Injecting the plaintext gate activation into the cipher forward pass restores 48% of refusals in Phi-4-mini, localizing the bypass to the routing interface. A second method, cipher contrast analysis, uses plain/cipher DLA differences to map the full cipher-sensitive routing circuit in O(3n) forward passes. Any encoding that defeats detection-layer pattern matching bypasses the policy regardless of whether deeper layers reconstruct the content.

Building cross-modal applications is challenging due to limited paired multi-modal data. Recent works have shown that leveraging a pre-trained multi-modal contrastive representation space enables cross-modal tasks to be learned from uni-modal data. This is based on the assumption that contrastive optimization makes embeddings from different modalities interchangeable. However, this assumption is under-explored due to the poorly understood geometry of the multi-modal contrastive space, where a modality gap exists. In our study, we provide a theoretical explanation of this space's geometry and introduce a three-step method, C^3 (Connect, Collapse, Corrupt), to bridge the modality gap, enhancing the interchangeability of embeddings. Our C^3 method significantly improves cross-modal learning from uni-modal data, achieving state-of-the-art results on zero-shot image / audio / video captioning and text-to-image generation.

A faithful and interpretable explanation of an AI model's behavior and internal structure is a high-level explanation that is human-intelligible but also consistent with the known, but often opaque low-level causal details of the model. We argue that the theory of causal abstraction provides the mathematical foundations for the desired kinds of model explanations. In causal abstraction analysis, we use interventions on model-internal states to rigorously assess whether an interpretable high-level causal model is a faithful description of an AI model. Our contributions in this area are: (1) We generalize causal abstraction to cyclic causal structures and typed high-level variables. (2) We show how multi-source interchange interventions can be used to conduct causal abstraction analyses. (3) We define a notion of approximate causal abstraction that allows us to assess the degree to which a high-level causal model is a causal abstraction of a lower-level one. (4) We prove constructive causal abstraction can be decomposed into three operations we refer to as marginalization, variable-merge, and value-merge. (5) We formalize the XAI methods of LIME, causal effect estimation, causal mediation analysis, iterated nullspace projection, and circuit-based explanations as special cases of causal abstraction analysis.

Recent years have witnessed major progress in joint audio-video generation. Existing models still suffer from limited per-modality fidelity, insufficient text-modality alignment and weak cross-modal synchronization. While reinforcement-learning post-training offers a promising remedy, directly adapting it to joint audio-video generation is challenging. Heterogeneous multimodal rewards entangle learning signals and complicate credit assignment. Joint optimization of two modality towers is computationally expensive given their divergent dynamics. Moreover, synchronization evaluation difficulty depends on paired samples, preventing fair reward comparisons. We propose AV-GRPO, a modality-anchored online diffusion RL framework, and 5DAV, a decoupled, difficulty-controllable training dataset. AV-GRPO includes three key modules: (1) modality-anchored rollouts to disentangle learning signals and stabilize difficulty; (2) trajectory-locked frozen-tower optimization to reduce cost and reassign credit; (3) adaptive objectives and perturbation strengths tailored to modality-specific dynamics. This converts coupled multimodal preference learning into unimodal subproblems for precise reward attribution and better synchronization. Our 5DAV dataset decouples samples across five dimensions for systematic training. Experiments on JavisBench and VABench demonstrate AV-GRPO outperforms LTX-2.3 in generation quality, semantic alignment and cross-modal synchronization under LoRA and full fine-tuning. Ablations confirm our designs. Code and data: https://github.com/zhiyuxu03/AV-GRPO

Recent advances in Video Foundation Models (VFMs) have revolutionized human-centric video synthesis, yet fine-grained and independent editing of subjects and scenes remains a critical challenge. Recent attempts to incorporate richer environment control through rigid 3D geometric compositions often encounter a stark trade-off between precise control and generative flexibility. Furthermore, the heavy 3D pre-processing still limits practical scalability. In this paper, we propose ONE-SHOT, a parameter-efficient framework for compositional human-environment video generation. Our key insight is to factorize the generative process into disentangled signals. Specifically, we introduce a canonical-space injection mechanism that decouples human dynamics from environmental cues via cross-attention. We also propose Dynamic-Grounded-RoPE, a novel positional embedding strategy that establishes spatial correspondences between disparate spatial domains without any heuristic 3D alignments. To support long-horizon synthesis, we introduce a Hybrid Context Integration mechanism to maintain subject and scene consistency across minute-level generations. Experiments demonstrate that our method significantly outperforms state-of-the-art methods, offering superior structural control and creative diversity for video synthesis. Our project has been available on: https://martayang.github.io/ONE-SHOT/.

Training large language models (LLMs) is often bottlenecked by extreme memory demands, with optimizer states dominating the footprint. Recent works mitigates this cost by projecting gradients into low-dimensional subspaces using sophisticated update strategies. In this paper, we analyze the dynamics of gradient space and its underlying subspaces. We find that while a small subspace captures most gradient energy, a significant portion still resides in the residual bulk; moreover, the influence of the core subspace diminishes over time and in deeper layers. We also observe that the gradient space exhibits near-flat curvature, calling for algorithms that explicitly account for this geometry. Motivated by these insights, we introduce a suite of randomized algorithms, GrassWalk and GrassJump, which exploit subspace and achieve state-of-the-art memory savings while improving performance on LLaMA-1B and LLaMA-7B pretraining.