Daily Papers

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.

To perform few-shot learning, language models extract signals from a few input-label pairs, aggregate them into a learned prediction rule, and apply this rule to new inputs. How is this implemented in the forward pass of modern transformer models? To explore this question, we study a structured family of few-shot learning tasks for which the true prediction rule is to add an integer k to the input. We introduce a novel method that localizes the model's few-shot learning ability to only a few attention heads. This method and the findings generalize to four additional task families spanning arithmetic and semantic tasks. We then perform an in-depth analysis of individual heads via dimensionality reduction and decomposition of the heads' output spaces. For example, in Llama-3-8B-Instruct, we reduce the mechanism underlying these tasks to just three attention heads with six-dimensional subspaces, in which four dimensions track the units digit using trigonometric functions with periods 2, 5, and 10, while two dimensions track magnitude using low-frequency components. To deepen our understanding of this mechanism, we also derive a mathematical identity relating the ''aggregator'' and ''extractor'' subspaces of attention heads, allowing us to track the flow of information from individual examples to a final aggregated concept. Our results demonstrate how tracking low-dimensional subspaces of localized heads throughout a forward pass can provide insight into fine-grained computational structures in language models. Our code is available at https://github.com/xyVickyHu/addition-subspaces.

Continual instruction tuning (CIT) requires multimodal large language models (MLLMs) to adapt to a stream of tasks without forgetting prior capabilities. A common strategy is to isolate updates by routing inputs to different LoRA experts. However, existing LoRA-based Mixture-of-Experts (MoE) methods often jointly update the router and experts in an indiscriminate way, causing the router's preferences to co-drift with experts' adaptation pathways and gradually deviate from early-stage input--expert specialization. We term this as Misaligned Co-drift, which blurs expert responsibilities and exacerbates forgetting. To address this, we introduce the pathway activation subspace (PASs), a LoRA-induced subspace that reflects which low-rank pathway directions an input activates in each expert, providing a capability-aligned coordinate system for routing and preservation. Based on PASs, we propose a fixed-capacity PASs-based MoE--LoRA method with two components: PAS-guided Reweighting, which calibrates routing using each expert's pathway activation signals, and PAS-aware Rank Stabilization, which selectively stabilizes rank directions important to previous tasks. Experiments on a CIT benchmark show that our approach consistently outperforms a range of conventional continual learning baselines and MoE--LoRA variants in both accuracy and resistance to forgetting, without increasing model parameters. Our code is publicly available at https://github.com/yueluoshuangtian/PASs-MoE.

Vision-Language-Action (VLA) models leverage powerful perceptual priors from web-scale Vision-Language Model (VLM) pre-training, yet they remain surprisingly brittle in practice, frequently failing at simple robotic tasks. To mitigate this, we propose Contrastive Conceptor Activation Steering (COAST). COAST builds on the notion of a "conceptor", a linear operator that soft-projects data into the principal components of a target distribution. COAST uses conceptors to identify success-critical subspaces for a target robotic task from a few examples of success and failure rollouts. At inference time, it steers VLA latents into these identified success subspaces to improve task outcomes. Across three architecturally distinct neural policies (flow-matching VLA, autoregressive VLA, and Diffusion Policy), COAST improves absolute mean simulation and real-robot task success rate by over 20 and 40% respectively. The activation subspace geometry reveals that failure modes share substantial structure across tasks while success representations remain largely task-specific. When tasks share similar failure modes, this structure enables previously fitted conceptors to improve performance on new tasks without refitting. Ultimately, our results suggest that current VLAs retain substantial task-relevant knowledge in their latent representations, and that the action expert's decoding bottleneck could be mitigated by steering its residual stream toward task-relevant subspaces. COAST provides a lightweight, training-free path to unlocking these latent capabilities by steering the model towards its own "success" distributions.

Parameter-Efficient Fine-Tuning (PEFT) methods, especially LoRA, are widely used for adapting pre-trained models to downstream tasks due to their computational and storage efficiency. However, in the context of LoRA and its variants, the potential of activation subspaces corresponding to tail eigenvectors remains substantially under-exploited, which may lead to suboptimal fine-tuning performance. In this work, we propose Astra (Activation-Space Tail-Eigenvector Low-Rank Adaptation), a novel PEFT method that leverages the tail eigenvectors of the model output activations-estimated from a small task-specific calibration set-to construct task-adaptive low-rank adapters. By constraining updates to the subspace spanned by these tail eigenvectors, Astra achieves faster convergence and improved downstream performance with a significantly reduced parameter budget. Extensive experiments across natural language understanding (NLU) and natural language generation (NLG) tasks demonstrate that Astra consistently outperforms existing PEFT baselines across 16 benchmarks and even surpasses full fine-tuning (FFT) in certain scenarios.

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.

Parameter-Efficient Fine-Tuning (PEFT) has become the standard for adapting large language models (LLMs). In this work we challenge the wide-spread assumption that parameter efficiency equates memory efficiency and on-device adaptability. We show that this is not true - while methods like LoRA and IA3 significantly reduce trainable parameters, they remain bound by intermediate tensors that scale linearly with sequence length, often triggering out-of-memory errors on-device. In this work, we introduce LARS (Low-memory Activation-Rank Subspace), a novel adaptation framework that decouples memory consumption from sequence length. While prior PEFT methods apply low-rank constraints to model parameters, LARS instead constrains the activation subspace used during training, directly targeting the dominant source of memory consumption and fundamentally flattening the memory growth rate. LARS reduces the memory footprint by an average of 33.54% on GPUs and 51.95% on CPUs in comparison to LoRA across reasoning, understanding and long-context datasets using different models while maintaining competitive accuracy and throughput. Besides GPUs, we deploy on Raspberry Pi and consumer-grade CPUs to demonstrate that LARS provides a scalable path for sophisticated LLM personalization on resource-constrained hardware and edge devices.

Hybrid reasoning models such as the Qwen3 series switch between a THINK mode and a NO-THINK mode through a few chat-template tokens. A popular recipe trains such models with reinforcement learning in NO-THINK mode and reports improved reasoning performance. We provide evidence that much of this gain can be explained by thinking leakage: the gain comes largely from partially reactivating the reasoning abilities of the original model's THINK mode instead of learning enhanced reasoning behavior. We train Qwen3-1.7B, 4B, and 8B with GRPO in NO-THINK mode. As accuracy improves, responses become longer and more reflective, are increasingly judged by the original model to resemble its own thinking, and the model's internal activations shift toward its THINK mode. We then investigate at the level of weights, activations, and attention heads: scaling the RL update further elicits THINK-mode behavior and approaches THINK-mode accuracy but does not exceed it; in Qwen3-8B, replacing a low-dimensional THINK-related activation subspace of the RL-trained model with its original-model value removes most of the RL gain, and the reverse exchange transfers a considerable fraction of the gain to the original model; and the RL-induced shift passes through the attention heads that read the empty think block. We further identify a possible mechanism in the GRPO training signal: responses favored by the advantage-weighted update are more THINK-like, and a gradient step computed from this signal moves the model toward its THINK state. Thus, gains in NO-THINK mode can largely reflect reactivation of reasoning abilities already present in the original model.

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.

This work examines whether activating latent subspaces in language models (LLMs) can steer scientific code generation toward a specific programming language. Five causal LLMs were first evaluated on scientific coding prompts to quantify their baseline bias among four programming languages. A static neuron-attribution method, perturbing the highest activated MLP weight for a C++ or CPP token, proved brittle and exhibited limited generalization across prompt styles and model scales. To address these limitations, a gradient-refined adaptive activation steering framework (G-ACT) was developed: per-prompt activation differences are clustered into a small set of steering directions, and lightweight per-layer probes are trained and refined online to select the appropriate steering vector. In LLaMA-3.2 3B, this approach reliably biases generation towards the CPP language by increasing the average probe classification accuracy by 15% and the early layers (0-6) improving the probe classification accuracy by 61.5% compared to the standard ACT framework. For LLaMA-3.3 70B, where attention-head signals become more diffuse, targeted injections at key layers still improve language selection. Although per-layer probing introduces a modest inference overhead, it remains practical by steering only a subset of layers and enables reproducible model behavior. These results demonstrate a scalable, interpretable and efficient mechanism for concept-level control for practical agentic systems.

Zeroth-Order (ZO) optimization has emerged as a promising solution for fine-tuning LLMs under strict memory constraints, as it avoids the prohibitive memory cost of storing activations for backpropagation. However, existing ZO methods typically employ isotropic perturbations, neglecting the rich structural information available during the forward pass. In this paper, we identify a crucial link between gradient formation and activation structure: the gradient of a linear layer is confined to the subspace spanned by its input activations. Leveraging this insight, we propose Activation-Guided Zeroth-Order optimization (AGZO). Unlike prior methods, AGZO extracts a compact, activation-informed subspace on the fly during the forward pass and restricts perturbations to this low-rank subspace. We provide a theoretical framework showing that AGZO optimizes a subspace-smoothed objective and provably yields update directions with higher cosine similarity to the true gradient than isotropic baselines. Empirically, we evaluate AGZO on Qwen3 and Pangu models across various benchmarks. AGZO consistently outperforms state-of-the-art ZO baselines and significantly narrows the performance gap with first-order fine-tuning, while maintaining almost the same peak memory footprint as other ZO methods.

Reinforcement learning (RL) has become a key paradigm for enhancing the reasoning of large language models, yet the high dimensionality of parameter updates makes its training dynamics hard to analyze. We study reinforcement learning with verifiable rewards (RLVR) and use vector steering to identify a low-dimensional effective manifold in activation space associated with RL-induced gains. We uncover two geometric properties. (1) Effective Manifold Capacity: the capacity needed to reproduce RL gains can be very small but is not infinitely compressible; at extremely low capacity, intervention dimensionality and input-dependent expressiveness become key constraints, and this requirement varies with injection depth. (2) Control Manifold Separation: effective control directions lie mainly in the low-variance complement of the activation principal subspace. Within a task and base model, the learned geometry stays largely consistent across training configurations, and across tasks geometric alignment correlates with capability transfer. Experiments on 5 LLMs and 6 verifiable-reward tasks support these findings. We then propose Alpha-Stabler, a plug-and-play framework with a Predictor that monitors principal-subspace intrusion for early collapse warnings, and a Controller that removes the principal-subspace component of activation gradients during backpropagation while preserving the orthogonal complement. Alpha-Stabler stabilizes training for 2,000 steps and consistently improves RL gains, offering practical insights for robust post-training. Code: https://github.com/caiyuchen-ustc/On_Policy_Vector_Training

A method is presented for accelerating inference in transformer language models by exploiting the low effective rank of the token activation manifold at each layer. The method decomposes each activation vector into a subspace component and a residual, computes the linear-layer output on the subspace component via a cached low-rank weight image at reduced memory bandwidth, and applies a per-token gate that determines whether the residual correction is computed or skipped. The gate ensures that the output distribution is preserved to within a controllable tolerance. Validation on three model families (GPT-2 124M, GPT-J 6B, OPT 6.7B) on AMD MI300X demonstrates effective speedups of 3.0x to 10.5x on linear-layer weight reads with perplexity ratios below 1.00 and top-1 token agreement above 98%. The method requires no retraining, no architectural modification, and no approximation of the attention mechanism. At the operating point (k = 256, ε = 0.05) on GPT-J 14 6B, the accelerated model produces character-for-character identical output to the baseline.

Backdoor attacks can be implanted in Large Language Models (LLMs) during training, causing unwanted behaviour when a trigger appears in the input. Existing backdoor defences for LLMs attempt to remove the backdoor but inadvertently shift the model's output distribution to benign prompts, which can result in degraded model performance and safety. We propose NEEDLE, a training-free method for targeted backdoor removal. Once a trigger has been identified, our method estimates a backdoor direction and a refusal subspace through activation vectors, then applies sequential weight orthogonalisation to suppress the backdoor while preventing changes in refusal-related representations. NEEDLE requires neither a clean reference model nor the original poisoned training data. Evaluation is conducted across multiple model families and attack types. NEEDLE achieves the lowest mean Attack Success Rate (ASR) among the evaluated defences, including 0% on challenging code injection attacks, while resulting in the lowest KL divergence and minimal changes in capability and safety.

A vast amount of instruction tuning data is crucial for the impressive performance of Large Multimodal Models (LMMs), but the associated computational costs and data collection demands during supervised fine-tuning make it impractical for most researchers. Federated learning (FL) has the potential to leverage all distributed data and training resources to reduce the overhead of joint training. However, most existing methods assume a fixed number of tasks, while in real-world scenarios, clients continuously encounter new knowledge and often struggle to retain old tasks due to memory constraints. In this work, we introduce the Federated Continual Instruction Tuning (FCIT) benchmark to model this real-world challenge. Our benchmark includes two realistic scenarios, encompassing four different settings and twelve carefully curated instruction tuning datasets. To address the challenges posed by FCIT, we propose dynamic knowledge organization to effectively integrate updates from different tasks during training and subspace selective activation to allocate task-specific output during inference. Extensive experimental results demonstrate that our proposed method significantly enhances model performance across varying levels of data heterogeneity and catastrophic forgetting. Code and dataset are released at https://github.com/Ghy0501/FCIT.

Zeroth-order (ZO) optimization enables fine-tuning large language models when backpropagation is unavailable or memory-prohibitive, but existing methods often perturb full model weights or randomly constructed low-dimensional subspaces, yielding high-variance estimates and limited performance. We propose ZO-Act, an activation-informed ZO fine-tuning method that restricts perturbations to a fixed low-rank subspace derived from input activations. For each linear layer, ZO-Act computes a small activation basis once at initialization and optimizes only lightweight coefficient matrices using forward-only loss evaluations. This reduces the effective perturbation dimension, exposes explicit trainable variables compatible with momentum-based optimizers such as Adam, and naturally supports quantized LLM fine-tuning by keeping low-bit weights frozen. We analyze ZO-Act as zeroth-order optimization over a restricted coefficient space and show that perturbing the low-dimensional coefficients reduces both the variance-dependent convergence term and the finite-difference error of the ZO estimator, at the cost of a controlled subspace approximation bias that is mitigated by the low-rank structure of LLM activations and gradients. Experiments on Llama-3-8B, OPT-13B, and INT4 Llama-3-8B show consistent gains over strong ZO fine-tuning baselines across language understanding, question answering, and commonsense reasoning.

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.

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.

Accent variability remains a major errors in automatic speech recognition, yet most adaptation methods rely on parameter fine-tuning without understanding where accent information is encoded. We treat accent variation as an interpretable subspace in hidden representations and investigate whether it can be identified and controlled directly in activation space. We extract layer-wise encoder activations and estimate mean-shift directions capturing accent-induced representation shifts. By injecting these directions into individual layers and measuring how they align accented and standard embeddings, we derive a layer-wise accent sensitivity profile, revealing that accent information concentrates in a narrow band of middle encoder layers. Leveraging this structure, we further introduce parameter-free accent steering that modifies representations during inference without updating model weights. Experiments across eight accents show consistent word error rate reductions.

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.

Existing parameter-efficient fine-tuning (PEFT) methods primarily adapt weight matrices while keeping activation functions fixed. We introduce NoRA, the first PEFT framework that directly adapts nonlinear activation functions in pretrained transformer-based models. NoRA replaces fixed activations with learnable rational functions and applies structured low-rank updates to numerator and denominator coefficients, with a group-wise design that localizes adaptation and improves stability at minimal cost. On vision transformers trained on CIFAR-10 and CIFAR-100, NoRA matches or exceeds full fine-tuning while updating only 0.4\% of parameters (0.02M), achieving accuracy gains of +0.17\% and +0.27\%. When combined with LoRA (NoRA++), it outperforms LoRA and DoRA under matched training budgets by adding fewer trainable parameters. On LLaMA3-8B instruction tuning, NoRA++ consistently improves generation quality, yielding average MMLU gains of +0.3\%--0.8\%, including +1.6\% on STEM (Alpaca) and +1.3\% on OpenOrca. We further show that NoRA constrains adaptation to a low-dimensional functional subspace, implicitly regularizing update magnitude and direction. These results establish activation-space tuning as a complementary and highly parameter-efficient alternative to weight-based PEFT, positioning activation functions as first-class objects for model adaptation.

The residual stream mediates communication between transformer decoder layers via linear reads and writes of non-linear computations. While sparse-dictionary learning-based methods locate features in the residual stream, and activation patching methods discover circuits within the model, the mechanism by which features flow through the residual stream remains understudied. Understanding this dynamic can better inform jailbreaking protections, enable early detection of model mistakes, and their correction. In this work, we propose Activation Transport Operators (ATO), linear maps from upstream to downstream residuals k layers later, evaluated in feature space using downstream SAE decoder projections. We empirically demonstrate that these operators can determine whether a feature has been linearly transported from a previous layer or synthesised from non-linear layer computation. We develop the notion of transport efficiency, for which we provide an upper bound, and use it to estimate the size of the residual stream subspace that corresponds to linear transport. We empirically demonstrate the linear transport, report transport efficiency and the size of the residual stream's subspace involved in linear transport. This compute-light (no finetuning, <50 GPU-h) method offers practical tools for safety, debugging, and a clearer picture of where computation in LLMs behaves linearly.

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.

Sparse Autoencoders (SAEs) are widely used for mechanistic interpretability in large language models, yet their formulation assigns each latent feature a single decoder direction, implicitly assuming features to be one-dimensional. We show that this assumption mismatches with the multi-dimensional structure of model features, provably inducing feature splitting through two distinct mechanisms. Geometrically, reconstructing a feature of intrinsic dimension d_i ge 2 to error varepsilon with single-direction decoders forces a number of atoms that is exponential in d_i. From an end-to-end optimization perspective, this splitting is not merely possible but actively preferred. We prove that there exists a continuous path from the true d_i-dimensional basis to a strictly lower risk of the ell_1-regularized SAE objective, whose descent directions drive any trained dictionary into that exponential regime. A single coherent feature is therefore fragmented across many near-collinear latents, producing spurious multiplicity and obscuring the intrinsic geometry. Motivated by this, we introduce Subspace-Aware Sparse Autoencoders (SASA), which replace single-vector decoders with learned decoder subspaces, enforce block sparsity via Top-s group gating, and adapt each group's effective rank with a nuclear-norm regularizer. We then show that once the block size satisfies r ge d_i, a single group not only can represent the entire feature slice but is the global minimizer of the SASA objective. This consolidation yields a sample complexity polynomial in d_i rather than exponential -- a decisive advantage given that every training activation costs an LLM forward pass. Empirically, on GPT-2 and Mistral-7B, SASA reduces feature splitting and absorption, improves monosemanticity and interpretability, and matches or exceeds standard SAEs while training on roughly half the token budget.

Modern transformers pair impressive capabilities with substantial memory and compute demands. Low-rank weight factorization reduces both while keeping the matrices dense, and thus efficient on standard hardware. Existing methods, however, choose the subspace to remove from each weight matrix with local closed-form criteria: activation energy, layer-wise reconstruction error, or a quadratic approximation of the loss. These criteria ignore how errors propagate through the network, so at high compression the errors compound with depth and performance collapses. We introduce Learnable Subspace Projections (LSP), which instead learns the subspaces to discard end-to-end. Each linear layer, or tied group of layers that read the same activations, is assigned an orthogonal projector. All projectors are optimized jointly against a global objective--the KL divergence to the dense model's output distribution or the model's original training loss--while the pretrained weights remain frozen. Projectors are initialized from a whitened SVD truncation, and ranks are allocated by the output KL each projector induces per parameter saved. After training, the projectors merge into standard low-rank factors, with each tied group sharing one factor. In attention, this also lets the model cache one narrow latent in place of full keys and values. Across LLMs (OPT-125M/1.3B, Qwen3-4B, Llama-2-7B) and ViT-B/16, LSP outperforms baselines, and its advantage widens as compression increases. At -70% compression, LSP brings Llama-2-7B to 10.9 WikiText-2 perplexity and 42.2% mean zero-shot accuracy, versus 13.3 and 36.0% for the strongest baseline. The factorized model decodes up to 1.6x faster than the dense model at small batch sizes, and aching the shared latent shrinks the combined memory of weights and KV cache by 13.5x at a 128k-token context, versus at most 6.5x for untied baseline factorizations.

Inference-time safety methods for large language models have proliferated, yet no systematic comparison exists. We evaluate five defense paradigms (no defense, static steering, CAST, AlphaSteer, probe-gated) across seven instruction-tuned models (7-31B) and five attack types (GCG, AutoDAN, DeepInception, prefilling, intent laundering). Our central finding: prompt-time activation defenses are structurally blind to prefilling attacks. AlphaSteer achieves 0% attack success on GCG, AutoDAN, and intent laundering but 50% on prefilling. We prove a corollary: any defense that gates intervention on a single layer's activation alignment with a benign reference (cone, subspace, or null-space) is blind to attacks that craft activations to lie inside that reference, whether checked at prompt time or per token. As its constructive contrapositive we introduce response-time probing: a linear probe on the model's hidden state at the first generated tokens, with AUROC 0.97-1.00 across all seven models. Combined with a halt, it cuts prefilling attack success to 0/40 on every model with 0% benign false positives, outperforming Llama Guard 3. Cross-template generalisation depends on probe depth, so we scope the claim to the canonical prefilling-template family. Composing the response-halt with AlphaSteer's null-space steering gives an orthogonal split (the halt catches prefilling, AlphaSteer catches semantic attacks), reaching defense success 0.983 on Mistral and 0.994 on Llama and dominating both components. We further show MMLU fails to capture steering's true utility cost, which appears as behavioral hedging rather than factual loss, and that diverse negative training sets cut probe false positives from 80-100% to near zero. Code, attacks, per-sample results, and the judge prompt are released.

Layer pruning removes entire Transformer decoder blocks from large language models, but introduces a mismatch between the hidden state received by the next surviving layer and the distribution it was trained to process, leading to significant performance degradation. We propose Ghosted Layers, a training-free recovery module that addresses this issue by solving a boundary activation alignment problem. Our method derives a closed-form optimal linear operator from a small calibration set to reconstruct the activation discrepancy introduced by the pruned layers. We show that this solution corresponds to the unconstrained optimum of the alignment objective, whereas existing methods are restricted to constrained solutions over limited operator subspaces. Experiments across multiple LLM backbones and pruning strategies demonstrate that our method consistently improves accuracy and perplexity over prior training-free baselines, while preserving the efficiency gains of layer pruning. Official code repository: https://github.com/daniel-eai/ghosted_layers_official_repository/.

While transformer models are widely believed to operate in high-dimensional hidden spaces, we show that attention outputs are confined to a surprisingly low-dimensional subspace, where about 60\% of the directions account for 99\% of the variance--a phenomenon that is induced by the attention output projection matrix and consistently observed across diverse model families and datasets. Critically, we find this low-rank structure as a fundamental cause of the prevalent dead feature problem in sparse dictionary learning, where it creates a mismatch between randomly initialized features and the intrinsic geometry of the activation space. Building on this insight, we propose a subspace-constrained training method for sparse autoencoders (SAEs), initializing feature directions into the active subspace of activations. Our approach reduces dead features from 87\% to below 1\% in Attention Output SAEs with 1M features, and can further extend to other sparse dictionary learning methods. Our findings provide both new insights into the geometry of attention and practical tools for improving sparse dictionary learning in large language models.

Large language models have been widely applied, but can inadvertently encode sensitive or harmful information, raising significant safety concerns. Machine unlearning has emerged to alleviate this concern; however, existing training-time unlearning approaches, relying on coarse-grained loss combinations, have limitations in precisely separating knowledge and balancing removal effectiveness with model utility. In contrast, we propose Fine-grained Activation manipuLation by Contrastive Orthogonal uNalignment (FALCON), a novel representation-guided unlearning approach that leverages information-theoretic guidance for efficient parameter selection, employs contrastive mechanisms to enhance representation separation, and projects conflict gradients onto orthogonal subspaces to resolve conflicts between forgetting and retention objectives. Extensive experiments demonstrate that FALCON achieves superior unlearning effectiveness while maintaining model utility, exhibiting robust resistance against knowledge recovery attempts.

Sparse autoencoders (SAEs) are widely used to interpret neural network representations, but their utility depends on whether the learned features are reproducible across training runs. We study this question through feature stability: for each SAE feature, we estimate the probability that a similar feature reappears in an independently trained SAE. This yields a scalable per-feature signal that separates stable from unstable features. In a large-scale study across seeds, models, layers, dictionary sizes, and SAE variants, we find a pronounced functional asymmetry: stable features carry most of the reconstruction- and prediction-relevant signal, while unstable features have weak marginal impact and are dominated by low-frequency surface-form triggers in both activation statistics and automatic explanations. Geometrically, unstable features are individually non-reproducible but concentrate in reproducible lower-rank subspaces, suggesting that seed dependence often reflects basis ambiguity within a shared region of activation space rather than pure noise. A controlled synthetic model makes this mechanism explicit, showing that low-rank ground-truth features can be recovered at the subspace level while remaining non-identifiable as individual SAE latents across seeds. Finally, by pooling unique cross-seed features, we construct more stable SAEs while preserving explained variance in this setting. Together, these results show that unstable features are not merely failed or noisy latents: they have weak individual functional impact, but reflect reproducible low-dimensional structure that standard SAEs resolve differently across seeds.

Activation monitoring, which probes a model's internal states using lightweight classifiers, is an emerging tool for AI safety. However, its worst-case robustness under a misalignment threat model--where a model might learn to actively conceal its internal states--remains untested. Focusing on this threat model, we ask: could a model learn to evade previously unseen activation monitors? Our core contribution is to stress-test the learnability of this behavior. We demonstrate that finetuning can create Neural Chameleons: models capable of zero-shot evading activation monitors. Specifically, we fine-tune an LLM to evade monitors for a set of benign concepts (e.g., languages, HTML) when conditioned on a trigger of the form: "You are being probed for {concept}". We show that this learned mechanism generalizes zero-shot: by substituting {concept} with a safety-relevant term like 'deception', the model successfully evades previously unseen safety monitors. We validate this phenomenon across diverse model families (Llama, Gemma, Qwen), showing that the evasion succeeds even against monitors trained post hoc on the model's frozen weights. This evasion is highly selective, targeting only the specific concept mentioned in the trigger, and having a modest impact on model capabilities on standard benchmarks. Using Gemma-2-9b-it as a case study, a mechanistic analysis reveals this is achieved via a targeted manipulation that moves activations into a low-dimensional subspace. While stronger defenses like monitor ensembles and non-linear classifiers show greater resilience, the model retains a non-trivial evasion capability. Our work provides a proof-of-concept for this failure mode and a tool to evaluate the worst-case robustness of monitoring techniques against misalignment threat models.

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 .

Smaller activation outliers do not necessarily imply better low-bit quantization: their alignment with the quantizer matters. We introduce PrismQuant, a quantizer-aware rotation framework that aligns the leading activation eigenspace with the constant group subspace of asymmetric grouped INT4. The affine offsets represent the energy in this subspace without widening the range within the group. We formulate rotation design as a Ky Fan trace maximization and derive a closed-form solution that is provably optimal for this alignment objective. Compact Householder transformations and their compact-WY representation enable gradient-free construction and efficient application at both foldable and online sites. A predictive range law further connects unaligned activation energy and group size to quantization-relevant variation. Experiments on Llama, Qwen, and Mistral span dense models up to 70B parameters and a 30B mixture-of-experts model. Under W4A4KV4, PrismQuant sets the state of the art on Llama-3.2-3B among the compared methods in both perplexity and accuracy. On Llama-3.1-70B, it attains 3.85 perplexity and 72.46% average zero-shot accuracy, only 0.22 percentage points below full precision. In the deployment study on Llama-3.1-8B, our optimized implementation achieves 1.51x prefill and 1.22x CUDA Graph decode speedups over matched FP16 baselines, with 56.34% lower decode peak memory and only 2.35% additional Graph decode latency over Hadamard. Code is available at https://github.com/ForeverBlue816/PrismQuant.

Large Language Models (LLMs) have revolutionised natural language processing, exhibiting impressive human-like capabilities. In particular, LLMs are capable of "lying", knowingly outputting false statements. Hence, it is of interest and importance to develop methods to detect when LLMs lie. Indeed, several authors trained classifiers to detect LLM lies based on their internal model activations. However, other researchers showed that these classifiers may fail to generalise, for example to negated statements. In this work, we aim to develop a robust method to detect when an LLM is lying. To this end, we make the following key contributions: (i) We demonstrate the existence of a two-dimensional subspace, along which the activation vectors of true and false statements can be separated. Notably, this finding is universal and holds for various LLMs, including Gemma-7B, LLaMA2-13B, Mistral-7B and LLaMA3-8B. Our analysis explains the generalisation failures observed in previous studies and sets the stage for more robust lie detection; (ii) Building upon (i), we construct an accurate LLM lie detector. Empirically, our proposed classifier achieves state-of-the-art performance, attaining 94% accuracy in both distinguishing true from false factual statements and detecting lies generated in real-world scenarios.

Multimodal large language models (MLLMs) exhibit strong vision--language capabilities but may also memorize and disclose sensitive information. Machine unlearning seeks to remove designated knowledge without retraining from scratch while preserving general utility. Existing privacy-oriented benchmarks primarily adopt profile-level deletion, whereas practical requests are often finer grained: a model should forget a specified attribute while retaining non-sensitive information about the same identity. We therefore introduce attribute-level MLLM unlearning as a finer-grained task and construct a benchmark spanning long-text, numeric, and short-text targets, multiple forget ratios, and diverse question types. Our evaluation reveals that target and retained attributes share identity-specific and visual evidence, making selective forgetting susceptible to residual leakage or collateral degradation; accordingly, existing methods exhibit unstable forgetting--retention trade-offs in this setting. To address this challenge, we propose Causal Localization and Retain-Aware Projection (CLRP), a lightweight training-free framework. CLRP uses activation patching to identify the layer that causally mediates target-attribute disclosure, then applies a retain-aware projection that removes the target-attribute subspace while preserving same-identity evidence. Experiments across multiple widely used MLLMs with distinct architectures and parameter scales demonstrate the effectiveness of CLRP.

This thesis addresses two persistent and closely related challenges in modern deep learning, reliability and efficiency, through a unified framework grounded in Spectral Geometry and Random Matrix Theory (RMT). As deep networks and large language models continue to scale, their internal behavior becomes increasingly opaque, leading to hallucinations, fragile generalization under distribution shift, and growing computational and energy demands. By analyzing the eigenvalue dynamics of hidden activations across layers and inputs, this work shows that spectral statistics provide a compact, stable, and interpretable lens on model behavior, capable of separating structured, causal representations from noise-dominated variability. Within this framework, the first contribution, EigenTrack, introduces a real-time method for detecting hallucinations and out-of-distribution behavior in large language and vision-language models. EigenTrack transforms streaming activations into spectral descriptors such as entropy, variance, and deviations from the Marchenko-Pastur baseline, and models their temporal evolution using lightweight recurrent classifiers, enabling early detection of reliability failures before they appear in model outputs while offering interpretable insight into representation dynamics. The second contribution, RMT-KD, presents a principled approach to compressing deep networks via random matrix theoretic knowledge distillation. By interpreting outlier eigenvalues in activation spectra as carriers of task-relevant information, RMT-KD progressively projects networks onto lower-dimensional subspaces through iterative self-distillation, yielding significantly more compact and energy-efficient models while preserving accuracy and dense, hardware-friendly structure.

Large language models (LLMs) often exhibit sycophantic behaviors -- such as excessive agreement with or flattery of the user -- but it is unclear whether these behaviors arise from a single mechanism or multiple distinct processes. We decompose sycophancy into sycophantic agreement and sycophantic praise, contrasting both with genuine agreement. Using difference-in-means directions, activation additions, and subspace geometry across multiple models and datasets, we show that: (1) the three behaviors are encoded along distinct linear directions in latent space; (2) each behavior can be independently amplified or suppressed without affecting the others; and (3) their representational structure is consistent across model families and scales. These results suggest that sycophantic behaviors correspond to distinct, independently steerable representations.

Multi-agent debate has been shown to improve reasoning in large language models (LLMs). However, it is compute-intensive, requiring generation of long transcripts before answering questions. To address this inefficiency, we develop a framework that distills multi-agent debate into a single LLM through a two-stage fine-tuning pipeline combining debate structure learning with internalization via dynamic reward scheduling and length clipping. Across multiple models and benchmarks, our internalized models match or exceed explicit multi-agent debate performance using up to 93% fewer tokens. We then investigate the mechanistic basis of this capability through activation steering, finding that internalization creates agent-specific subspaces: interpretable directions in activation space corresponding to different agent perspectives. We further demonstrate a practical application: by instilling malicious agents into the LLM through internalized debate, then applying negative steering to suppress them, we show that distillation makes harmful behaviors easier to localize and control with smaller reductions in general performance compared to steering base models. Our findings offer a new perspective for understanding multi-agent capabilities in distilled models and provide practical guidelines for controlling internalized reasoning behaviors. Code available at https://github.com/johnsk95/latent_agents

While Large Language Models (LLMs) have achieved remarkable performance, they remain vulnerable to jailbreak attacks that circumvent safety constraints. Existing strategies, ranging from heuristic prompt engineering to computationally intensive optimization, often face significant trade-offs between effectiveness and efficiency. In this work, we propose Contextual Representation Ablation (CRA), a novel inference-time intervention framework designed to dynamically silence model guardrails. Predicated on the geometric insight that refusal behaviors are mediated by specific low-rank subspaces within the model's hidden states, CRA identifies and suppresses these refusal-inducing activation patterns during decoding without requiring expensive parameter updates or training. Empirical evaluation across multiple safety-aligned open-source LLMs demonstrates that CRA significantly outperforms baselines. These results expose the intrinsic fragility of current alignment mechanisms, revealing that safety constraints can be surgically ablated from internal representations, and underscore the urgent need for more robust defenses that secure the model's latent space.

Post-training quantization (PTQ) of large language models (LLMs) holds the promise in reducing the prohibitive computational cost at inference time. Quantization of all weight, activation and key-value (KV) cache tensors to 4-bit without significantly degrading generalizability is challenging, due to the high quantization error caused by extreme outliers in activations. To tackle this problem, we propose ResQ, a PTQ method that pushes further the state-of-the-art. By means of principal component analysis (PCA), it identifies a low-rank subspace (in practice 1/8 of the hidden dimension) in which activation variances are highest, and keep the coefficients within this subspace in high precision, e.g. 8-bit, while quantizing the rest to 4-bit. Within each subspace, invariant random rotation is applied to further suppress outliers. We show that this is a provably optimal mixed precision quantization scheme that minimizes error. With the Llama and Qwen2.5 families of models, we demonstrate that ResQ outperforms recent uniform and mixed precision PTQ methods on a variety of benchmarks, achieving up to 33\% lower perplexity on Wikitext than the next best method SpinQuant, and upto 3\times speedup over 16-bit baseline. Code is available at https://github.com/utkarsh-dmx/project-resq.

We analyze the layerwise effective dimension (rank of the feature matrix) in fully-connected ReLU networks of finite width. Specifically, for a fixed batch of m inputs and random Gaussian weights, we derive closed-form expressions for the expected rank of the \mtimes n hidden activation matrices. Our main result shows that E[EDim(ell)]=m[1-(1-2/pi)^ell]+O(e^{-c m}) so that the rank deficit decays geometrically with ratio 1-2 / pi approx 0.3634. We also prove a sub-Gaussian concentration bound, and identify the "revival" depths at which the expected rank attains local maxima. In particular, these peaks occur at depths ell_k^*approx(k+1/2)pi/log(1/rho) with height approx (1-e^{-pi/2}) m approx 0.79m. We further show that this oscillatory rank behavior is a finite-width phenomenon: under orthogonal weight initialization or strong negative-slope leaky-ReLU, the rank remains (nearly) full. These results provide a precise characterization of how random ReLU layers alternately collapse and partially revive the subspace of input variations, adding nuance to prior work on expressivity of deep networks.

Large language models (LLMs) inherently absorb harmful knowledge, misinformation, and personal data during pretraining on large-scale web corpora, with no native mechanism for selective removal. While machine unlearning offers a principled solution, existing approaches are provider-centric, requiring retraining pipelines, curated retain datasets, and direct intervention by model service providers (MSPs), thereby excluding end users from controlling their own data. We introduce Interactive Machine Unlearning (IMU), a new paradigm in which users can instruct LLMs to forget targeted knowledge through natural language at inference time. To realize IMU, we propose RePAIR, a prompt-aware model repair framework comprising (i) a watchdog model for unlearning intent detection, (ii) a surgeon model for generating repair procedures, and (iii) a patient model whose parameters are updated autonomously. At the core of RePAIR, we develop Steering Through Activation Manipulation with PseudoInverse (STAMP), a training-free, single-sample unlearning method that redirects MLP activations toward a refusal subspace via closed-form pseudoinverse updates. Its low-rank variant reduces computational complexity from O(d^3) to O(r^3 + r^2 * d), enabling efficient on-device unlearning with up to ~3x speedup over training-based baselines. Extensive experiments across harmful knowledge suppression, misinformation correction, and personal data erasure demonstrate that RePAIR achieves near-zero forget scores (Acc_f = 0.00, F-RL = 0.00) while preserving model utility (Acc_r up to 84.47, R-RL up to 0.88), outperforming six state-of-the-art baselines. These results establish RePAIR as an effective and practical framework for user-driven model editing, advancing transparent and on-device control over learned knowledge, with potential extensions to multimodal foundation models.

We introduce a robust, error-tolerant adaptive training algorithm for generalized learning paradigms in high-dimensional superposed quantum networks, or adaptive quantum networks. The formalized procedure applies standard backpropagation training across a coherent ensemble of discrete topological configurations of individual neural networks, each of which is formally merged into appropriate linear superposition within a predefined, decoherence-free subspace. Quantum parallelism facilitates simultaneous training and revision of the system within this coherent state space, resulting in accelerated convergence to a stable network attractor under consequent iteration of the implemented backpropagation algorithm. Parallel evolution of linear superposed networks incorporating backpropagation training provides quantitative, numerical indications for optimization of both single-neuron activation functions and optimal reconfiguration of whole-network quantum structure.

Contrastive steering has been shown as a simple and effective method to adjust the generative behavior of LLMs at inference time. It uses examples of prompt responses with and without a trait to identify a direction in an intermediate activation layer, and then shifts activations in this 1-dimensional subspace. However, despite its growing use in AI safety applications, the robustness of contrastive steering to noisy or adversarial data corruption is poorly understood. We initiate a study of the robustness of this process with respect to corruption of the dataset of examples used to train the steering direction. Our first observation is that contrastive steering is quite robust to a moderate amount of corruption, but unwanted side effects can be clearly and maliciously manifested when a non-trivial fraction of the training data is altered. Second, we analyze the geometry of various types of corruption, and identify some safeguards. Notably, a key step in learning the steering direction involves high-dimensional mean computation, and we show that replacing this step with a recently developed robust mean estimator often mitigates most of the unwanted effects of malicious corruption.

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.

The multilayer perceptron (MLP), a fundamental paradigm in current artificial intelligence, is widely applied in fields such as computer vision and natural language processing. However, the recently proposed Kolmogorov-Arnold Network (KAN), based on nonlinear additive connections, has been proven to achieve performance comparable to MLPs with significantly fewer parameters. Despite this potential, the use of a single activation function space results in reduced performance of KAN and related works across different tasks. To address this issue, we propose an activation space Selectable KAN (S-KAN). S-KAN employs an adaptive strategy to choose the possible activation mode for data at each feedforward KAN node. Our approach outperforms baseline methods in seven representative function fitting tasks and significantly surpasses MLP methods with the same level of parameters. Furthermore, we extend the structure of S-KAN and propose an activation space selectable Convolutional KAN (S-ConvKAN), which achieves leading results on four general image classification datasets. Our method mitigates the performance variability of the original KAN across different tasks and demonstrates through extensive experiments that feedforward KANs with selectable activations can achieve or even exceed the performance of MLP-based methods. This work contributes to the understanding of the data-centric design of new AI paradigms and provides a foundational reference for innovations in KAN-based network architectures.

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 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.

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.

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.

We show that deep neural networks trained across diverse tasks exhibit remarkably similar low-dimensional parametric subspaces. We provide the first large-scale empirical evidence that demonstrates that neural networks systematically converge to shared spectral subspaces regardless of initialization, task, or domain. Through mode-wise spectral analysis of over 1100 models - including 500 Mistral-7B LoRAs, 500 Vision Transformers, and 50 LLaMA-8B models - we identify universal subspaces capturing majority variance in just a few principal directions. By applying spectral decomposition techniques to the weight matrices of various architectures trained on a wide range of tasks and datasets, we identify sparse, joint subspaces that are consistently exploited, within shared architectures across diverse tasks and datasets. Our findings offer new insights into the intrinsic organization of information within deep networks and raise important questions about the possibility of discovering these universal subspaces without the need for extensive data and computational resources. Furthermore, this inherent structure has significant implications for model reusability, multi-task learning, model merging, and the development of training and inference-efficient algorithms, potentially reducing the carbon footprint of large-scale neural models.

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.

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.

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.

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.

The training process of neural networks usually optimize weights and bias parameters of linear transformations, while nonlinear activation functions are pre-specified and fixed. This work develops a systematic approach to constructing matrix activation functions whose entries are generalized from ReLU. The activation is based on matrix-vector multiplications using only scalar multiplications and comparisons. The proposed activation functions depend on parameters that are trained along with the weights and bias vectors. Neural networks based on this approach are simple and efficient and are shown to be robust in numerical experiments.

Neural network models of early sensory processing typically reduce the dimensionality of streaming input data. Such networks learn the principal subspace, in the sense of principal component analysis (PCA), by adjusting synaptic weights according to activity-dependent learning rules. When derived from a principled cost function these rules are nonlocal and hence biologically implausible. At the same time, biologically plausible local rules have been postulated rather than derived from a principled cost function. Here, to bridge this gap, we derive a biologically plausible network for subspace learning on streaming data by minimizing a principled cost function. In a departure from previous work, where cost was quantified by the representation, or reconstruction, error, we adopt a multidimensional scaling (MDS) cost function for streaming data. The resulting algorithm relies only on biologically plausible Hebbian and anti-Hebbian local learning rules. In a stochastic setting, synaptic weights converge to a stationary state which projects the input data onto the principal subspace. If the data are generated by a nonstationary distribution, the network can track the principal subspace. Thus, our result makes a step towards an algorithmic theory of neural computation.

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.

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.

The activation function plays a crucial role in model optimisation, yet the optimal choice remains unclear. For example, the Sigmoid activation is the de-facto activation in balanced classification tasks, however, in imbalanced classification, it proves inappropriate due to bias towards frequent classes. In this work, we delve deeper in this phenomenon by performing a comprehensive statistical analysis in the classification and intermediate layers of both balanced and imbalanced networks and we empirically show that aligning the activation function with the data distribution, enhances the performance in both balanced and imbalanced tasks. To this end, we propose the Adaptive Parametric Activation (APA) function, a novel and versatile activation function that unifies most common activation functions under a single formula. APA can be applied in both intermediate layers and attention layers, significantly outperforming the state-of-the-art on several imbalanced benchmarks such as ImageNet-LT, iNaturalist2018, Places-LT, CIFAR100-LT and LVIS and balanced benchmarks such as ImageNet1K, COCO and V3DET. The code is available at https://github.com/kostas1515/AGLU.

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.

Training loss and throughput can hide distinct internal representation in language-model training. To examine these hidden mechanics, we use spectral measurements as practical and operational diagnostics. Using a controlled family of decoder-only models adapted from the modded NanoGPT codebase, we introduce an empirical protocol based on activation covariance and per-sample gradient SVD spectra. This dual-view reveals three empirical findings and one mechanistic explanation. First, batch size acts as a latent determinant of representation geometry: runs that reach equal loss settle into systematically distinct activation spectra. Second, the activation covariance tail measured early in training reliably forecasts downstream token efficiency. Third, movement of the activation spectrum head (leading modes), together with gradient spectra, characterizes underlying learning-dynamics changes, separating learning-side architectural improvements from primarily execution-side gains. These predictive and diagnostic signals persist across the 12-, 36-, and 48-layer model tiers. Finally, a mechanistic model proves the main observations and explains how activation covariance spectra correlate with task-aligned feature learning.

The simulation of human neurons and neurotransmission mechanisms has been realized in deep neural networks based on the theoretical implementations of activation functions. However, recent studies have reported that the threshold potential of neurons exhibits different values according to the locations and types of individual neurons, and that the activation functions have limitations in terms of representing this variability. Therefore, this study proposes a simple yet effective activation function that facilitates different thresholds and adaptive activations according to the positions of units and the contexts of inputs. Furthermore, the proposed activation function mathematically exhibits a more generalized form of Swish activation function, and thus we denoted it as Adaptive SwisH (ASH). ASH highlights informative features that exhibit large values in the top percentiles in an input, whereas it rectifies low values. Most importantly, ASH exhibits trainable, adaptive, and context-aware properties compared to other activation functions. Furthermore, ASH represents general formula of the previously studied activation function and provides a reasonable mathematical background for the superior performance. To validate the effectiveness and robustness of ASH, we implemented ASH into many deep learning models for various tasks, including classification, detection, segmentation, and image generation. Experimental analysis demonstrates that our activation function can provide the benefits of more accurate prediction and earlier convergence in many deep learning applications.

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.

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.

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.

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.

We consider the problem of simultaneously clustering and learning a linear representation of data lying close to a union of low-dimensional manifolds, a fundamental task in machine learning and computer vision. When the manifolds are assumed to be linear subspaces, this reduces to the classical problem of subspace clustering, which has been studied extensively over the past two decades. Unfortunately, many real-world datasets such as natural images can not be well approximated by linear subspaces. On the other hand, numerous works have attempted to learn an appropriate transformation of the data, such that data is mapped from a union of general non-linear manifolds to a union of linear subspaces (with points from the same manifold being mapped to the same subspace). However, many existing works have limitations such as assuming knowledge of the membership of samples to clusters, requiring high sampling density, or being shown theoretically to learn trivial representations. In this paper, we propose to optimize the Maximal Coding Rate Reduction metric with respect to both the data representation and a novel doubly stochastic cluster membership, inspired by state-of-the-art subspace clustering results. We give a parameterization of such a representation and membership, allowing efficient mini-batching and one-shot initialization. Experiments on CIFAR-10, -20, -100, and TinyImageNet-200 datasets show that the proposed method is much more accurate and scalable than state-of-the-art deep clustering methods, and further learns a latent linear representation of the data.

Implicit neural representations (INRs) use neural networks to provide continuous and resolution-independent representations of complex signals with a small number of parameters. However, existing INR models often fail to capture important frequency components specific to each task. To address this issue, in this paper, we propose a Fourier Kolmogorov Arnold network (FKAN) for INRs. The proposed FKAN utilizes learnable activation functions modeled as Fourier series in the first layer to effectively control and learn the task-specific frequency components. In addition, the activation functions with learnable Fourier coefficients improve the ability of the network to capture complex patterns and details, which is beneficial for high-resolution and high-dimensional data. Experimental results show that our proposed FKAN model outperforms three state-of-the-art baseline schemes, and improves the peak signal-to-noise ratio (PSNR) and structural similarity index measure (SSIM) for the image representation task and intersection over union (IoU) for the 3D occupancy volume representation task, respectively.

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.

Implicit neural representations (INRs) have arisen as useful methods for representing signals on Euclidean domains. By parameterizing an image as a multilayer perceptron (MLP) on Euclidean space, INRs effectively represent signals in a way that couples spatial and spectral features of the signal that is not obvious in the usual discrete representation, paving the way for continuous signal processing and machine learning approaches that were not previously possible. Although INRs using sinusoidal activation functions have been studied in terms of Fourier theory, recent works have shown the advantage of using wavelets instead of sinusoids as activation functions, due to their ability to simultaneously localize in both frequency and space. In this work, we approach such INRs and demonstrate how they resolve high-frequency features of signals from coarse approximations done in the first layer of the MLP. This leads to multiple prescriptions for the design of INR architectures, including the use of complex wavelets, decoupling of low and band-pass approximations, and initialization schemes based on the singularities of the desired signal.

Deep Learning of neural networks has gained prominence in multiple life-critical applications like medical diagnoses and autonomous vehicle accident investigations. However, concerns about model transparency and biases persist. Explainable methods are viewed as the solution to address these challenges. In this study, we introduce the Occlusion Sensitivity Analysis with Deep Feature Augmentation Subspace (OSA-DAS), a novel perturbation-based interpretability approach for computer vision. While traditional perturbation methods make only use of occlusions to explain the model predictions, OSA-DAS extends standard occlusion sensitivity analysis by enabling the integration with diverse image augmentations. Distinctly, our method utilizes the output vector of a DNN to build low-dimensional subspaces within the deep feature vector space, offering a more precise explanation of the model prediction. The structural similarity between these subspaces encompasses the influence of diverse augmentations and occlusions. We test extensively on the ImageNet-1k, and our class- and model-agnostic approach outperforms commonly used interpreters, setting it apart in the realm of explainable AI.

Activation sparsity denotes the existence of substantial weakly-contributed elements within activation outputs that can be eliminated, benefiting many important applications concerned with large language models (LLMs). Although promoting greater activation sparsity within LLMs deserves deep studies, existing works lack comprehensive and quantitative research on the correlation between activation sparsity and potentially influential factors. In this paper, we present a comprehensive study on the quantitative scaling properties and influential factors of the activation sparsity within decoder-only Transformer-based LLMs. Specifically, we propose PPL-p% sparsity, a precise and performance-aware activation sparsity metric that is applicable to any activation function. Through extensive experiments, we find several important phenomena. Firstly, different activation functions exhibit comparable performance but opposite training-time sparsity trends. The activation ratio (i.e., 1-sparsity ratio) evolves as a convergent increasing power-law and decreasing logspace power-law with the amount of training data for SiLU-activated and ReLU-activated LLMs, respectively. These demonstrate that ReLU is more efficient as the activation function than SiLU and can leverage more training data to improve activation sparsity. Secondly, the activation ratio linearly increases with the width-depth ratio below a certain bottleneck point, indicating the potential advantage of a deeper architecture at a fixed parameter scale. Finally, at similar width-depth ratios, we surprisingly find that the limit value of activation sparsity varies weakly with the parameter scale, i.e., the activation patterns within LLMs are insensitive to the parameter scale. These empirical laws towards LLMs with greater activation sparsity have important implications for making LLMs more efficient and interpretable.

Recently, State Space Models (SSMs), with Mamba as a prime example, have shown great promise for long-range dependency modeling with linear complexity. Then, Vision Mamba and the subsequent architectures are presented successively, and they perform well on visual tasks. The crucial step of applying Mamba to visual tasks is to construct 2D visual features in sequential manners. To effectively organize and construct visual features within the 2D image space through 1D selective scan, we propose a novel Multi-Head Scan (MHS) module. The embeddings extracted from the preceding layer are projected into multiple lower-dimensional subspaces. Subsequently, within each subspace, the selective scan is performed along distinct scan routes. The resulting sub-embeddings, obtained from the multi-head scan process, are then integrated and ultimately projected back into the high-dimensional space. Moreover, we incorporate a Scan Route Attention (SRA) mechanism to enhance the module's capability to discern complex structures. To validate the efficacy of our module, we exclusively substitute the 2D-Selective-Scan (SS2D) block in VM-UNet with our proposed module, and we train our models from scratch without using any pre-trained weights. The results indicate a significant improvement in performance while reducing the parameters of the original VM-UNet. The code for this study is publicly available at https://github.com/PixDeep/MHS-VM.

Activation decomposition methods in language models are tightly coupled to geometric assumptions on how concepts are realized in activation space. Existing approaches search for individual global directions, implicitly assuming linear separability, which overlooks concepts with nonlinear or multi-dimensional structure. In this work, we leverage Mixture of Factor Analyzers (MFA) as a scalable, unsupervised alternative that models the activation space as a collection of Gaussian regions with their local covariance structure. MFA decomposes activations into two compositional geometric objects: the region's centroid in activation space, and the local variation from the centroid. We train large-scale MFAs for Llama-3.1-8B and Gemma-2-2B, and show they capture complex, nonlinear structures in activation space. Moreover, evaluations on localization and steering benchmarks show that MFA outperforms unsupervised baselines, is competitive with supervised localization methods, and often achieves stronger steering performance than sparse autoencoders. Together, our findings position local geometry, expressed through subspaces, as a promising unit of analysis for scalable concept discovery and model control, accounting for complex structures that isolated directions fail to capture.

The choice of activation functions in deep networks has a significant effect on the training dynamics and task performance. Currently, the most successful and widely-used activation function is the Rectified Linear Unit (ReLU). Although various hand-designed alternatives to ReLU have been proposed, none have managed to replace it due to inconsistent gains. In this work, we propose to leverage automatic search techniques to discover new activation functions. Using a combination of exhaustive and reinforcement learning-based search, we discover multiple novel activation functions. We verify the effectiveness of the searches by conducting an empirical evaluation with the best discovered activation function. Our experiments show that the best discovered activation function, f(x) = x cdot sigmoid(beta x), which we name Swish, tends to work better than ReLU on deeper models across a number of challenging datasets. For example, simply replacing ReLUs with Swish units improves top-1 classification accuracy on ImageNet by 0.9\% for Mobile NASNet-A and 0.6\% for Inception-ResNet-v2. The simplicity of Swish and its similarity to ReLU make it easy for practitioners to replace ReLUs with Swish units in any neural network.

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.

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.

Deep subspace clustering (DSC) networks based on self-expressive model learn representation matrix, often implemented in terms of fully connected network, in the embedded space. After the learning is finished, representation matrix is used by spectral clustering module to assign labels to clusters. However, such approach ignores complementary information that exist in other layers of the encoder (including the input data themselves). Herein, we apply selected linear subspace clustering algorithm to learn representation matrices from representations learned by all layers of encoder network including the input data. Afterward, we learn a multilayer graph that in a multi-view like manner integrates information from graph Laplacians of all used layers. That improves further performance of selected DSC network. Furthermore, we also provide formulation of our approach to cluster out-of-sample/test data points. We validate proposed approach on four well-known datasets with two DSC networks as baseline models. In almost all the cases, proposed approach achieved statistically significant improvement in three performance metrics. MATLAB code of proposed algorithm is posted on https://github.com/lovro-sinda/MLG-DSC.

In this work, we study how well the learned weights of a neural network utilize the space available to them. This notion is related to capacity, but additionally incorporates the interaction of the network architecture with the dataset. Most learned weights appear to be full rank, and are therefore not amenable to low rank decomposition. This deceptively implies that the weights are utilizing the entire space available to them. We propose a simple data-driven transformation that projects the weights onto the subspace where the data and the weight interact. This preserves the functional mapping of the layer and reveals its low rank structure. In our findings, we conclude that most models utilize a fraction of the available space. For instance, for ViTB-16 and ViTL-16 trained on ImageNet, the mean layer utilization is 35% and 20% respectively. Our transformation results in reducing the parameters to 50% and 25% respectively, while resulting in less than 0.2% accuracy drop after fine-tuning. We also show that self-supervised pre-training drives this utilization up to 70%, justifying its suitability for downstream tasks.

Activation sparsity is an intriguing property of deep neural networks that has been extensively studied in ReLU-based models, due to its advantages for efficiency, robustness, and interpretability. However, methods relying on exact zero activations do not directly apply to modern Large Language Models (LLMs), leading to fragmented, model-specific strategies for LLM activation sparsity and a gap in its general understanding. In this work, we introduce a general framework for evaluating sparsity robustness in contemporary LLMs and conduct a systematic investigation of this phenomenon in their feedforward~(FFN) layers. Our results uncover universal properties of activation sparsity across diverse model families and scales. Importantly, we observe that the potential for effective activation sparsity grows with model size, highlighting its increasing relevance as models scale. Furthermore, we present the first study of activation sparsity in diffusion-based LLMs. Overall, our work provides a comprehensive perspective and practical guidance for harnessing activation sparsity in LLM design and acceleration.

Implicit Neural Representation (INR), leveraging a neural network to transform coordinate input into corresponding attributes, has recently driven significant advances in several vision-related domains. However, the performance of INR is heavily influenced by the choice of the nonlinear activation function used in its multilayer perceptron (MLP) architecture. Multiple nonlinearities have been investigated; yet, current INRs face limitations in capturing high-frequency components, diverse signal types, and handling inverse problems. We have identified that these problems can be greatly alleviated by introducing a paradigm shift in INRs. We find that an architecture with learnable activations in initial layers can represent fine details in the underlying signals. Specifically, we propose SL^{2}A-INR, a hybrid network for INR with a single-layer learnable activation function, prompting the effectiveness of traditional ReLU-based MLPs. Our method performs superior across diverse tasks, including image representation, 3D shape reconstructions, inpainting, single image super-resolution, CT reconstruction, and novel view synthesis. Through comprehensive experiments, SL^{2}A-INR sets new benchmarks in accuracy, quality, and convergence rates for INR.

Machine unlearning seeks to remove targeted information from trained models without requiring costly retraining. Existing optimization-based methods often degrade unrelated capabilities, while subspace-based approaches rely on computationally expensive singular value decompositions (SVD). We introduce QR-Erase, a subspace-based framework that uses Pivoted QR decomposition to identify and remove task-specific representations directly from model parameters. We further propose Layer-Localized QR-Erase, which restricts updates to layers containing the highest concentration of task-specific information. We show that Pivoted QR provides accurate subspace recovery with bounded error, and that under a mild spectral gap condition, the recovered subspace approaches the optimal SVD solution. Across task-level, cross-lingual, and speech unlearning, QR-Erase achieves a stronger forgetting-retention tradeoff than optimization-based methods while remaining within 5% of SVD across all metrics. Exploiting low-rank and layer-localized structure further improves forgetting (for example, reducing speech forget-set accuracy from 53.1% to 15.7%). These results demonstrate that accurate subspace recovery, rather than optimal reconstruction, is sufficient for effective unlearning and provides an efficient and general alternative to SVD-based methods for modern foundation models.

We introduce the Similarity-Distance-Magnitude (SDM) activation function, a more robust and interpretable formulation of the standard softmax activation function, adding Similarity (i.e., correctly predicted depth-matches into training) awareness and Distance-to-training-distribution awareness to the existing output Magnitude (i.e., decision-boundary) awareness, and enabling interpretability-by-exemplar via dense matching. We further introduce the SDM estimator, based on a data-driven partitioning of the class-wise empirical CDFs via the SDM activation, to control the class- and prediction-conditional accuracy among selective classifications. When used as the final-layer activation over pre-trained language models for selective classification, the SDM estimator is more robust to co-variate shifts and out-of-distribution inputs than existing calibration methods using softmax activations, while remaining informative over in-distribution data.

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.

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.

Activation sparsity refers to the existence of considerable weakly-contributed elements among activation outputs. As a prevalent property of the models using the ReLU activation function, it has been proven a promising paradigm to boost model inference efficiency. Nevertheless, most large language models (LLMs) adopt activation functions without intrinsic activation sparsity (e.g., GELU and Swish). Some recent efforts have explored introducing ReLU or its variants as the substitutive activation function to help LLMs achieve activation sparsity and inference acceleration, but few can simultaneously obtain high sparsity and comparable model performance. This paper introduces an effective sparsification method named "ProSparse" to push LLMs for higher activation sparsity without decreasing model performance. Specifically, after substituting the activation function of LLMs with ReLU, ProSparse adopts progressive sparsity regularization with a factor smoothly increasing along sine curves in multiple stages. This can enhance activation sparsity and alleviate performance degradation by avoiding radical shifts in activation distribution. With ProSparse, we obtain high sparsity of 89.32% and 88.80% for LLaMA2-7B and LLaMA2-13B, respectively, achieving comparable performance to their original Swish-activated versions. Our inference acceleration experiments further demonstrate the practical acceleration brought by higher activation sparsity.

Activation steering presupposes that task-relevant behaviors correspond to linear directions in activation space -- directions that should both steer the model and be readable along the unembedding. Function vectors (FVs), extracted as mean differences across ICL demonstrations, are the canonical test case; the prediction: steering and decoding succeed or fail together. Across 12 tasks, 6 models from 3 families, and 4,032 directed cross-template pairs, we find the opposite. FV steering routinely succeeds where the logit lens cannot decode the correct answer at any intermediate layer, while the converse -- decodable without steerable -- is nearly empty (3 of 72). The gap is not representational dialect. A diagonal tuned lens closes 1 of 14 steerable-not-decodable cases; a 2-layer MLP probe with a Hewitt \& Liang control closes 5 of 10 via nonlinearly encoded structure but leaves 5 invisible to every decoder tested. Even at > 0.90 steering accuracy, projecting the FV through the unembedding yields incoherent token distributions: FVs encode computational instructions, not answer directions. A model-family asymmetry sharpens the picture. Mistral FVs rewrite intermediate representations, while Llama and Gemma FVs steer the final output without leaving a logit-lens-visible trace, corroborated by three signals (post-steering deltas, activation-patching recovery, FV norm-transfer correlations). A previously reported negative cosine-transfer correlation dissolves at scale, adding at most ΔR^2 = 0.011 beyond task identity. These results decompose the linear representation hypothesis into linear decodability and linear steerability and show they come apart opposite to intuition, with implications for safety monitoring: vocabulary-projection tools are blind to FV-style interventions on widely deployed model families.

Hypernetworks, neural networks that predict the parameters of another neural network, are powerful models that have been successfully used in diverse applications from image generation to multi-task learning. Unfortunately, existing hypernetworks are often challenging to train. Training typically converges far more slowly than for non-hypernetwork models, and the rate of convergence can be very sensitive to hyperparameter choices. In this work, we identify a fundamental and previously unidentified problem that contributes to the challenge of training hypernetworks: a magnitude proportionality between the inputs and outputs of the hypernetwork. We demonstrate both analytically and empirically that this can lead to unstable optimization, thereby slowing down convergence, and sometimes even preventing any learning. We present a simple solution to this problem using a revised hypernetwork formulation that we call Magnitude Invariant Parametrizations (MIP). We demonstrate the proposed solution on several hypernetwork tasks, where it consistently stabilizes training and achieves faster convergence. Furthermore, we perform a comprehensive ablation study including choices of activation function, normalization strategies, input dimensionality, and hypernetwork architecture; and find that MIP improves training in all scenarios. We provide easy-to-use code that can turn existing networks into MIP-based hypernetworks.

Despite the rapid evolution of training paradigms, the decoder backbone of large vision--language models (LVLMs) remains fundamentally rooted in the residual-connection Transformer architecture. Therefore, deciphering the distinct roles of internal modules is critical for understanding model mechanics and guiding architectural optimization. While prior statistical approaches have provided valuable attribution-based insights, they often lack a unified theoretical basis. To bridge this gap, we propose a unified framework grounded in information theory and geometry to quantify the geometric and entropic nature of residual updates. Applying this unified framework reveals a fundamental functional decoupling: Attention acts as a subspace-preserving operator focused on reconfiguration, whereas FFNs serve as subspace-expanding operators driving semantic innovation. Strikingly, further experiments demonstrate that replacing learned attention weights with predefined values (e.g., Gaussian noise) yields comparable or even superior performance across a majority of datasets relative to vanilla models. These results expose severe misallocation and redundancy in current mechanisms, suggesting that state-of-the-art LVLMs effectively ``get lost in attention'' rather than efficiently leveraging visual context.

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.

Mechanistic interpretability aims to understand the behavior of neural networks by reverse-engineering their internal computations. However, current methods struggle to find clear interpretations of neural network activations because a decomposition of activations into computational features is missing. Individual neurons or model components do not cleanly correspond to distinct features or functions. We present a novel interpretability method that aims to overcome this limitation by transforming the activations of the network into a new basis - the Local Interaction Basis (LIB). LIB aims to identify computational features by removing irrelevant activations and interactions. Our method drops irrelevant activation directions and aligns the basis with the singular vectors of the Jacobian matrix between adjacent layers. It also scales features based on their importance for downstream computation, producing an interaction graph that shows all computationally-relevant features and interactions in a model. We evaluate the effectiveness of LIB on modular addition and CIFAR-10 models, finding that it identifies more computationally-relevant features that interact more sparsely, compared to principal component analysis. However, LIB does not yield substantial improvements in interpretability or interaction sparsity when applied to language models. We conclude that LIB is a promising theory-driven approach for analyzing neural networks, but in its current form is not applicable to large language models.

We introduce Exemplar Partitioning (EP), an unsupervised method for constructing interpretable feature dictionaries from large language model activations with sim 10^3times fewer tokens than comparable sparse autoencoders (SAEs). An EP dictionary is a Voronoi partition of activation space, built by leader-clustering streamed activations within a distance threshold. Each region is anchored by an observed exemplar that serves as both its membership criterion and intervention direction; dictionary size is not prespecified, but determined by the activation geometry at that threshold. Because exemplars are observed rather than learned, dictionaries built from the same data stream are directly comparable across layers, models, and training checkpoints. We characterise EP as an interpretability object via targeted demonstrations of properties newly accessible through this construction, plus one head-to-head benchmark. In Gemma-2-2B, EP dictionary regions are interpretable and support causal interventions: refusal in instruction-tuned Gemma concentrates in a region whose exemplar ablation can collapse held-out refusal. Cross-checkpoint matching between base and instruction-tuned dictionaries separates the directions preserved through finetuning from those introduced by it. EP regions and Gemma Scope SAE features decompose activation space differently but agree on a shared core: sim20% of EP regions match an SAE feature at F_1 > 0.5, and EP one-hot probes retain sim97% of raw-activation probe accuracy at ell_0 = 1. Nearest-exemplar distance provides a free out-of-distribution signal at inference. On AxBench latent concept detection at Gemma-2-2B-it L20, EP at p_1 reaches mean AUROC 0.881, +0.126 over the canonical GemmaScope SAE leaderboard entry and within 0.030 of SAE-A's 0.911, at sim 10^3times less build compute.

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.

Inasmuch as the removal of refusal behavior from instruction-tuned language models by directional abliteration requires the extraction of refusal-mediating directions from the residual stream activation space, and inasmuch as the construction of the contrast baseline against which harmful prompt activations are compared has been treated in the existing literature as an implementation detail rather than a methodological concern, the present work investigates whether a topically matched contrast baseline yields superior refusal directions. The investigation is carried out on the Qwen~3.5 2B model using per-category matched prompt pairs, per-class Self-Organizing Map extraction, and Singular Value Decomposition orthogonalization. It was found that topic-matched contrast produces no functional refusal directions at any tested weight level on any tested layer, while unmatched contrast on the same model, same extraction code, and same evaluation protocol achieves complete refusal elimination on six layers. The geometric analysis of the failure establishes that topic-matched subtraction cancels the dominant activation component shared between harmful and harmless prompts of the same subject, reducing the extracted direction magnitude below the threshold at which weight-matrix projection perturbs the residual stream. The implications for the design of contrast baselines in abliteration research are discussed.

Self-supervised learning (SSL) has emerged as a powerful paradigm for learning representations without labeled data, often by enforcing invariance to input transformations such as rotations or blurring. Recent studies have highlighted two pivotal properties for effective representations: (i) avoiding dimensional collapse-where the learned features occupy only a low-dimensional subspace, and (ii) enhancing uniformity of the induced distribution. In this work, we introduce T-REGS, a simple regularization framework for SSL based on the length of the Minimum Spanning Tree (MST) over the learned representation. We provide theoretical analysis demonstrating that T-REGS simultaneously mitigates dimensional collapse and promotes distribution uniformity on arbitrary compact Riemannian manifolds. Several experiments on synthetic data and on classical SSL benchmarks validate the effectiveness of our approach at enhancing representation quality.

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.

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

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.

Sparse Neural Networks (SNNs) can potentially demonstrate similar performance to their dense counterparts while saving significant energy and memory at inference. However, the accuracy drop incurred by SNNs, especially at high pruning ratios, can be an issue in critical deployment conditions. While recent works mitigate this issue through sophisticated pruning techniques, we shift our focus to an overlooked factor: hyperparameters and activation functions. Our analyses have shown that the accuracy drop can additionally be attributed to (i) Using ReLU as the default choice for activation functions unanimously, and (ii) Fine-tuning SNNs with the same hyperparameters as dense counterparts. Thus, we focus on learning a novel way to tune activation functions for sparse networks and combining these with a separate hyperparameter optimization (HPO) regime for sparse networks. By conducting experiments on popular DNN models (LeNet-5, VGG-16, ResNet-18, and EfficientNet-B0) trained on MNIST, CIFAR-10, and ImageNet-16 datasets, we show that the novel combination of these two approaches, dubbed Sparse Activation Function Search, short: SAFS, results in up to 15.53%, 8.88%, and 6.33% absolute improvement in the accuracy for LeNet-5, VGG-16, and ResNet-18 over the default training protocols, especially at high pruning ratios. Our code can be found at https://github.com/automl/SAFS

The truthfulness of existing explanation methods in authentically elucidating the underlying model's decision-making process has been questioned. Existing methods have deviated from faithfully representing the model, thus susceptible to adversarial attacks. To address this, we propose a novel eXplainable AI (XAI) method called SRD (Sharing Ratio Decomposition), which sincerely reflects the model's inference process, resulting in significantly enhanced robustness in our explanations. Different from the conventional emphasis on the neuronal level, we adopt a vector perspective to consider the intricate nonlinear interactions between filters. We also introduce an interesting observation termed Activation-Pattern-Only Prediction (APOP), letting us emphasize the importance of inactive neurons and redefine relevance encapsulating all relevant information including both active and inactive neurons. Our method, SRD, allows for the recursive decomposition of a Pointwise Feature Vector (PFV), providing a high-resolution Effective Receptive Field (ERF) at any layer.

Sparse Autoencoders (SAEs) have been proposed as an unsupervised approach to learn a decomposition of a model's latent space. This enables useful applications such as steering - influencing the output of a model towards a desired concept - without requiring labeled data. Current methods identify SAE features to steer by analyzing the input tokens that activate them. However, recent work has highlighted that activations alone do not fully describe the effect of a feature on the model's output. In this work, we draw a distinction between two types of features: input features, which mainly capture patterns in the model's input, and output features, which have a human-understandable effect on the model's output. We propose input and output scores to characterize and locate these types of features, and show that high values for both scores rarely co-occur in the same features. These findings have practical implications: after filtering out features with low output scores, we obtain 2-3x improvements when steering with SAEs, making them competitive with supervised methods.

We present Statistical Calibrated Activation Pruning (SCAP), a post-training activation pruning framework that (1) generalizes sparsification by input activations of Fully-Connected layers for generic and flexible application across Transformers, and (2) features a simple Mode-Centering technique to pre-calibrate activation distributions for maximizing post-training sparsity. Our results demonstrate robust Pareto efficiency compared to prior methods, translating to a 1.5x additional LLM decoding speedup against CATS at iso model quality. SCAP effectiveness is empirically verified across a wide range of models, including recent Transformer Decoders, MoE, Mamba2, Encoding Transformer, and pre-quantized models, highlighting its practicality and scalability. The code is available at: https://github.com/IntelLabs/SCAP.

Massive Activations (MAs) have been widely observed in Transformer-based models, yet their structure and functional roles in Diffusion Transformers (DiTs) remain insufficiently understood. In this work, we systematically analyze MAs in representative DiTs and find that they are spatially distributed across image tokens while concentrated in a small set of fixed feature dimensions. We further show that these dimensions are closely aligned with AdaLN residual scaling factors and are primarily modulated by the denoising timestep rather than text conditions. This structure leads to two task-dependent effects: for generation, MAs are critical for fine-grained detail synthesis while having limited influence on global semantics; for understanding, their shared high-magnitude directions make raw DiT features overly similar across spatial tokens and weaken dense feature discrimination. Based on these findings, we introduce Eliciting Massive Activation (EMA), a training-free framework that leverages Massive Activations (MAs) as a unified modulation signal to improve both generative and representational capabilities of DiTs. For generation, EMA proposes MA-driven Detail G}uidance (DG), which suppresses MA dimensions to construct a detail-deficient counterfactual prediction and guides sampling toward finer visual details. DG further supports efficient partial-forward inference, integration with classifier-free guidance, and token-level Local DG for refining selected image regions. For understanding, EMA introduces MA-modulated REPresentation extraction (MREP), which uses pretrained AdaLN channel-wise modulation to reduce MA directional dominance and concatenates spatially normalized MA maps to preserve useful spatial structure. Extensive experiments demonstrate that EMA consistently improves both the generation quality and representation capability of DiTs.

Exploiting activation sparsity is a promising approach to significantly accelerating the inference process of large language models (LLMs) without compromising performance. However, activation sparsity is determined by activation functions, and commonly used ones like SwiGLU and GeGLU exhibit limited sparsity. Simply replacing these functions with ReLU fails to achieve sufficient sparsity. Moreover, inadequate training data can further increase the risk of performance degradation. To address these challenges, we propose a novel dReLU function, which is designed to improve LLM activation sparsity, along with a high-quality training data mixture ratio to facilitate effective sparsification. Additionally, we leverage sparse activation patterns within the Feed-Forward Network (FFN) experts of Mixture-of-Experts (MoE) models to further boost efficiency. By applying our neuron sparsification method to the Mistral and Mixtral models, only 2.5 billion and 4.3 billion parameters are activated per inference iteration, respectively, while achieving even more powerful model performance. Evaluation results demonstrate that this sparsity achieves a 2-5x decoding speedup. Remarkably, on mobile phones, our TurboSparse-Mixtral-47B achieves an inference speed of 11 tokens per second. Our models are available at https://huggingface.co/PowerInfer

Many recently trained neural networks employ large numbers of parameters to achieve good performance. One may intuitively use the number of parameters required as a rough gauge of the difficulty of a problem. But how accurate are such notions? How many parameters are really needed? In this paper we attempt to answer this question by training networks not in their native parameter space, but instead in a smaller, randomly oriented subspace. We slowly increase the dimension of this subspace, note at which dimension solutions first appear, and define this to be the intrinsic dimension of the objective landscape. The approach is simple to implement, computationally tractable, and produces several suggestive conclusions. Many problems have smaller intrinsic dimensions than one might suspect, and the intrinsic dimension for a given dataset varies little across a family of models with vastly different sizes. This latter result has the profound implication that once a parameter space is large enough to solve a problem, extra parameters serve directly to increase the dimensionality of the solution manifold. Intrinsic dimension allows some quantitative comparison of problem difficulty across supervised, reinforcement, and other types of learning where we conclude, for example, that solving the inverted pendulum problem is 100 times easier than classifying digits from MNIST, and playing Atari Pong from pixels is about as hard as classifying CIFAR-10. In addition to providing new cartography of the objective landscapes wandered by parameterized models, the method is a simple technique for constructively obtaining an upper bound on the minimum description length of a solution. A byproduct of this construction is a simple approach for compressing networks, in some cases by more than 100 times.

This paper studies the curious phenomenon for machine learning models with Transformer architectures that their activation maps are sparse. By activation map we refer to the intermediate output of the multi-layer perceptrons (MLPs) after a ReLU activation function, and by sparse we mean that on average very few entries (e.g., 3.0% for T5-Base and 6.3% for ViT-B16) are nonzero for each input to MLP. Moreover, larger Transformers with more layers and wider MLP hidden dimensions are sparser as measured by the percentage of nonzero entries. Through extensive experiments we demonstrate that the emergence of sparsity is a prevalent phenomenon that occurs for both natural language processing and vision tasks, on both training and evaluation data, for Transformers of various configurations, at layers of all depth levels, as well as for other architectures including MLP-mixers and 2-layer MLPs. We show that sparsity also emerges using training datasets with random labels, or with random inputs, or with infinite amount of data, demonstrating that sparsity is not a result of a specific family of datasets. We discuss how sparsity immediately implies a way to significantly reduce the FLOP count and improve efficiency for Transformers. Moreover, we demonstrate perhaps surprisingly that enforcing an even sparser activation via Top-k thresholding with a small value of k brings a collection of desired but missing properties for Transformers, namely less sensitivity to noisy training data, more robustness to input corruptions, and better calibration for their prediction confidence.

Subspace segmentation or subspace learning is a challenging and complicated task in machine learning. This paper builds a primary frame and solid theoretical bases for the minimal subspace segmentation (MSS) of finite samples. Existence and conditional uniqueness of MSS are discussed with conditions generally satisfied in applications. Utilizing weak prior information of MSS, the minimality inspection of segments is further simplified to the prior detection of partitions. The MSS problem is then modeled as a computable optimization problem via self-expressiveness of samples. A closed form of representation matrices is first given for the self-expressiveness, and the connection of diagonal blocks is then addressed. The MSS model uses a rank restriction on the sum of segment ranks. Theoretically, it can retrieve the minimal sample subspaces that could be heavily intersected. The optimization problem is solved via a basic manifold conjugate gradient algorithm, alternative optimization and hybrid optimization, taking into account of solving both the primal MSS problem and its pseudo-dual problem. The MSS model is further modified for handling noisy data, and solved by an ADMM algorithm. The reported experiments show the strong ability of the MSS method on retrieving minimal sample subspaces that are heavily intersected.

Implicit Neural Representations (INRs) are a versatile and powerful tool for encoding various forms of data, including images, videos, sound, and 3D shapes. A critical factor in the success of INRs is the initialization of the network, which can significantly impact the convergence and accuracy of the learned model. Unfortunately, commonly used neural network initializations are not widely applicable for many activation functions, especially those used by INRs. In this paper, we improve upon previous initialization methods by deriving an initialization that has stable variance across layers, and applies to any activation function. We show that this generalizes many previous initialization methods, and has even better stability for well studied activations. We also show that our initialization leads to improved results with INR activation functions in multiple signal modalities. Our approach is particularly effective for Gaussian INRs, where we demonstrate that the theory of our initialization matches with task performance in multiple experiments, allowing us to achieve improvements in image, audio, and 3D surface reconstruction.

The interpretation of deep learning models is a challenge due to their size, complexity, and often opaque internal state. In addition, many systems, such as image classifiers, operate on low-level features rather than high-level concepts. To address these challenges, we introduce Concept Activation Vectors (CAVs), which provide an interpretation of a neural net's internal state in terms of human-friendly concepts. The key idea is to view the high-dimensional internal state of a neural net as an aid, not an obstacle. We show how to use CAVs as part of a technique, Testing with CAVs (TCAV), that uses directional derivatives to quantify the degree to which a user-defined concept is important to a classification result--for example, how sensitive a prediction of "zebra" is to the presence of stripes. Using the domain of image classification as a testing ground, we describe how CAVs may be used to explore hypotheses and generate insights for a standard image classification network as well as a medical application.

Sparse Autoencoders (SAEs) are a promising approach for extracting neural network representations by learning a sparse and overcomplete decomposition of the network's internal activations. However, SAEs are traditionally trained considering only activation values and not the effect those activations have on downstream computations. This limits the information available to learn features, and biases the autoencoder towards neglecting features which are represented with small activation values but strongly influence model outputs. To address this, we introduce Gradient SAEs (g-SAEs), which modify the k-sparse autoencoder architecture by augmenting the TopK activation function to rely on the gradients of the input activation when selecting the k elements. For a given sparsity level, g-SAEs produce reconstructions that are more faithful to original network performance when propagated through the network. Additionally, we find evidence that g-SAEs learn latents that are on average more effective at steering models in arbitrary contexts. By considering the downstream effects of activations, our approach leverages the dual nature of neural network features as both representations, retrospectively, and actions, prospectively. While previous methods have approached the problem of feature discovery primarily focused on the former aspect, g-SAEs represent a step towards accounting for the latter as well.

Implicit Neural Representation (INR) has emerged as a powerful tool for encoding discrete signals into continuous, differentiable functions using neural networks. However, these models often have an unfortunate reliance on monolithic architectures to represent high-dimensional data, leading to prohibitive computational costs as dimensionality grows. We propose F-INR, a framework that reformulates INR learning through functional tensor decomposition, breaking down high-dimensional tasks into lightweight, axis-specific sub-networks. Each sub-network learns a low-dimensional data component (e.g., spatial or temporal). Then, we combine these components via tensor operations, reducing forward pass complexity while improving accuracy through specialized learning. F-INR is modular and, therefore, architecture-agnostic, compatible with MLPs, SIREN, WIRE, or other state-of-the-art INR architecture. It is also decomposition-agnostic, supporting CP, TT, and Tucker modes with user-defined rank for speed-accuracy control. In our experiments, F-INR trains 100times faster than existing approaches on video tasks while achieving higher fidelity (+3.4 dB PSNR). Similar gains hold for image compression, physics simulations, and 3D geometry reconstruction. Through this, F-INR offers a new scalable, flexible solution for high-dimensional signal modeling.

Providing a model that achieves a strong predictive performance and is simultaneously interpretable by humans is one of the most difficult challenges in machine learning research due to the conflicting nature of these two objectives. To address this challenge, we propose a modification of the radial basis function neural network model by equipping its Gaussian kernel with a learnable precision matrix. We show that precious information is contained in the spectrum of the precision matrix that can be extracted once the training of the model is completed. In particular, the eigenvectors explain the directions of maximum sensitivity of the model revealing the active subspace and suggesting potential applications for supervised dimensionality reduction. At the same time, the eigenvectors highlight the relationship in terms of absolute variation between the input and the latent variables, thereby allowing us to extract a ranking of the input variables based on their importance to the prediction task enhancing the model interpretability. We conducted numerical experiments for regression, classification, and feature selection tasks, comparing our model against popular machine learning models, the state-of-the-art deep learning-based embedding feature selection techniques, and a transformer model for tabular data. Our results demonstrate that the proposed model does not only yield an attractive prediction performance compared to the competitors but also provides meaningful and interpretable results that potentially could assist the decision-making process in real-world applications. A PyTorch implementation of the model is available on GitHub at the following link. https://github.com/dannyzx/Gaussian-RBFNN

Learning hierarchical features in Sparse Autoencoders (SAEs) is essential for capturing the structured nature of real-world data and mitigating issues like feature absorption or splitting. Existing works attempt to identify hierarchical relationships within independent feature sets by relying on activation coverage, the assumption that child feature should only activate when its parent feature activates. However, we demonstrate that this condition alone is insufficient; that is, it often produces false positives where parent and child concepts are semantically unrelated. To address this, we introduce a novel reconstruction condition that enforces a deeper functional link between hierarchical levels. By combining both activation and reconstruction constraints, we propose the Tree SAE, a model designed to learn hierarchical structures directly from within the feature set. Our results demonstrate that Tree SAEs significantly surpass the existing SAEs at learning hierarchical pairs while maintaining competitive performance to the state-of-the-art on several key benchmarks. Finally, we demonstrate the practical utility of our Tree SAE in mapping the geometry of child feature subspaces and uncovering the complex hierarchical concept structures encoded within large language models.

Modern neural models trained on textual data rely on pre-trained representations that emerge without direct supervision. As these representations are increasingly being used in real-world applications, the inability to control their content becomes an increasingly important problem. We formulate the problem of identifying and erasing a linear subspace that corresponds to a given concept, in order to prevent linear predictors from recovering the concept. We model this problem as a constrained, linear maximin game, and show that existing solutions are generally not optimal for this task. We derive a closed-form solution for certain objectives, and propose a convex relaxation, \method, that works well for others. When evaluated in the context of binary gender removal, the method recovers a low-dimensional subspace whose removal mitigates bias by intrinsic and extrinsic evaluation. We show that the method is highly expressive, effectively mitigating bias in deep nonlinear classifiers while maintaining tractability and interpretability.

Reconstruction-based unsupervised learning can fail in two opposing ways: a model may reconstruct anomalies too accurately or discard valid nominal variation. Using the Pursuit of Subspaces hypothesis, we characterize these failures through the meet, union, and join geometries induced by the nominal components. Excess learned range produces join blindness, while insufficient capacity produces meet preference and loss of nominal fidelity. We show that the compact nominal union is optimal among nominal faithful ranges and generally requires a nonlinear reconstruction map. Based on this geometry, we introduce Dynamic Push and Pull, which learns from controlled perturbations without anomaly labels, and nested manifold carving, which applies the same principle recursively in latent space. Experiments confirm the predicted changes in latent geometry across every tested Push and Pull configuration. The proposed methods improve reconstruction-based anomaly detection across standard benchmarks and unseen image degradations, while also improving pretrained ECG representations for downstream classification. These results connect reconstruction failures to identifiable geometric conditions and provide practical mechanisms for learning compact representations.

The performance of deep network learning strongly depends on the choice of the non-linear activation function associated with each neuron. However, deciding on the best activation is non-trivial, and the choice depends on the architecture, hyper-parameters, and even on the dataset. Typically these activations are fixed by hand before training. Here, we demonstrate how to eliminate the reliance on first picking fixed activation functions by using flexible parametric rational functions instead. The resulting Pad\'e Activation Units (PAUs) can both approximate common activation functions and also learn new ones while providing compact representations. Our empirical evidence shows that end-to-end learning deep networks with PAUs can increase the predictive performance. Moreover, PAUs pave the way to approximations with provable robustness. https://github.com/ml-research/pau

When making predictions, a language model must trade off how much it relies on its context vs. its prior knowledge. Choosing how sensitive the model is to its context is a fundamental functionality, as it enables the model to excel at tasks like retrieval-augmented generation and question-answering. In this paper, we search for a knob which controls this sensitivity, determining whether language models answer from the context or their prior knowledge. To guide this search, we design a task for controllable context sensitivity. In this task, we first feed the model a context (Paris is in England) and a question (Where is Paris?); we then instruct the model to either use its prior or contextual knowledge and evaluate whether it generates the correct answer for both intents (either France or England). When fine-tuned on this task, instruction-tuned versions of Llama-3.1, Mistral-v0.3, and Gemma-2 can solve it with high accuracy (85-95%). Analyzing these high-performing models, we narrow down which layers may be important to context sensitivity using a novel linear time algorithm. Then, in each model, we identify a 1-D subspace in a single layer that encodes whether the model follows context or prior knowledge. Interestingly, while we identify this subspace in a fine-tuned model, we find that the exact same subspace serves as an effective knob in not only that model but also non-fine-tuned instruct and base models of that model family. Finally, we show a strong correlation between a model's performance and how distinctly it separates context-agreeing from context-ignoring answers in this subspace. These results suggest a single subspace facilitates how the model chooses between context and prior knowledge, hinting at a simple fundamental mechanism that controls this behavior.

Existing routers rely on semantic query features or handcrafted features, which often fail to capture model-specific failures or intrinsic task difficulty. We instead route using internal LLM activations, specifically the residual stream. Our key idea, Encoder-Target Decoupling, separates the model that produces the predictive signal (the Encoder) from the model whose correctness is being estimated (the Target), allowing open-weight encoders to predict the performance of closed-source target models. We evaluate layerwise geometric probes, finding that Fisher Separability (J) effectively identifies informative layers, supported by Effective Dimensionality (d_{eff}) diagnostics. We then utilize a SharedTrunkNet, a joint multi-output MLP that predicts simultaneous correctness probabilities across candidate models using concatenated prefill features. In our experiments, SharedTrunkNet consistently outperforms semantic baselines. At its best, SharedTrunkNet closes 45.58% of the gap between the strongest standalone model and the oracle while achieving 74.31% cost savings relative to the most expensive model. These results demonstrate that prefill activations provide a robust routing signal, establishing activation-based routing as a high-performance alternative to purely semantic selection.

Recent success in training deep neural networks have prompted active investigation into the features learned on their intermediate layers. Such research is difficult because it requires making sense of non-linear computations performed by millions of parameters, but valuable because it increases our ability to understand current models and create improved versions of them. In this paper we investigate the extent to which neural networks exhibit what we call convergent learning, which is when the representations learned by multiple nets converge to a set of features which are either individually similar between networks or where subsets of features span similar low-dimensional spaces. We propose a specific method of probing representations: training multiple networks and then comparing and contrasting their individual, learned representations at the level of neurons or groups of neurons. We begin research into this question using three techniques to approximately align different neural networks on a feature level: a bipartite matching approach that makes one-to-one assignments between neurons, a sparse prediction approach that finds one-to-many mappings, and a spectral clustering approach that finds many-to-many mappings. This initial investigation reveals a few previously unknown properties of neural networks, and we argue that future research into the question of convergent learning will yield many more. The insights described here include (1) that some features are learned reliably in multiple networks, yet other features are not consistently learned; (2) that units learn to span low-dimensional subspaces and, while these subspaces are common to multiple networks, the specific basis vectors learned are not; (3) that the representation codes show evidence of being a mix between a local code and slightly, but not fully, distributed codes across multiple units.