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38 papers in the last four weeks, up 111% on the four weeks before. 0.5% of all new papers.

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Latest papers 279

Oct 1, 2026cs.CV

Continual Concept Erasure in Diffusion Models by Suppressing Cross-Edit Interference

Concept erasure removes copyright-protected, privacy-sensitive, or otherwise undesirable concepts from pretrained text-to-image diffusion models to support content governance and compliance. As erasure requests arrive over time, models must remove new targets without undoing prior erasures. Existing methods do not constrain interference across edits: residual perturbations outside the retain set interact and accumulate, degrading unrelated generations and sometimes collapsing previously erased targets into noise. We propose CEASE (Continual Erasure via Adaptive Subspace Editing), a training-free method that imposes two subspace constraints on a closed-form solver. CEASE adds the token representation of the shared replacement to the solver's invariance matrix and, when interference is detected, projects the current update onto the orthogonal complement of dominant output directions extracted from cumulative past updates. A closed-form decomposition attributes the accumulated interference to repeated activation of the shared replacement and overlap between successive update directions, showing that the two constraints suppress these respective sources. Across continual erasure of celebrities, artistic styles, and instances, CEASE achieves the most consistent erase-preserve trade-off, while existing methods either degrade general generation or insufficiently erase targets.

Yongliang Wu, Haori Lu, Jinqi Luo +3

University of Illinois Urbana-Champaign · University of Pennsylvania · National University of Singapore

Concept Erasure Text-To-Image Diffusion Models Diffusion Models Content Moderation Interference Subspace

Oct 1, 2026cs.AI

FedLore: Communication and Memory Efficient Federated Learning via Shared Gradient Low-Rank Projection

Federated training of foundation models is constrained by client memory and communication costs. LoRA-based methods reduce these costs through low-rank adapters, but their fixed rank budget can limit adaptation. Gradient low-rank optimization offers greater flexibility, yet independently chosen client subspaces create a problem we term \emph{subspace fragmentation}: local projections interact with data heterogeneity to bias aggregated directions, while aggregation can increase update rank and communication cost. Thus, accurate local gradient compression need not preserve global descent. We propose \texttt{FedLore}, which shares a low-rank optimization basis within each round and refreshes it across rounds. The shared basis enables exact aggregation in low-rank coordinates and eliminates the identified projection bias. Subspace refresh allows the accumulated model update to exceed the per-round rank budget. We characterize the aggregation bias and establish an

O(T^{-1/2})

stationarity bound for the projected-SGD variant under a global-gradient coverage condition and standard smoothness and variance assumptions, with bounded gradient heterogeneity. Experiments on vision and language tasks, including federated pre-training, show that \texttt{FedLore} outperforms the evaluated low-rank adapter baselines and matches or exceeds full-parameter training, while reducing communication and optimizer-state memory.

Junkang Liu

Tianjin University

Federated Learning Fedavg Low-Rank Structure Subspace Fragmentation Projection

Oct 1, 2026cs.CL

HeadEdit: Calibrating Language Model Behavior Through the Frozen Unembedding Matrix

Alignment does not eliminate behavioral errors in language models. Models may still refuse benign requests, call unnecessary tools, or yield to false user claims. Current methods mitigate such errors as a computation problem, and rarely explore if the desired behavior is already encoded in the model's representation. Motivated by the observation that behavior-relevant information remains linearly decodable from the final hidden state even when the resulting logits produce the undesired behavior, we introduce HeadEdit, a gradient-free method that calibrates model behavior through the unembedding matrix. HeadEdit extracts a low-rank behavioral subspace from paired completions and uses each prompt's coordinates within it to generate a vocabulary-wide correction, thereby implementing implicitly adaptive steering without manually specified target tokens or parameter updates. HeadEdit improves all nine experimental settings across three tasks and three model families, with negligible inference overhead and no systematic loss of general capabilities. It also reveals a connection to gradient-based alignment. HeadEdit's low-dimensional representation partly predicts how preference tuning changes output logits on unseen prompts. The subspace learned from the model can also be reused after tuning, improving performance without re-extracting or retuning. These results show that HeadEdit provides a practical, lightweight, and interpretable way to calibrate model behavior through the unembedding matrix.

Zirui He, Haiyan Zhao, Jingyu Hu +5

New Jersey Institute of Technology · University of Bristol · Kuaishou Technology +1

Large Language Model Alignment Tuning Subspace Matrix

Sep 30, 2026cs.LG

Signal-Noise Factorization Isolates Nuisance Variation into Removable Subspaces

Recent theoretical work identified fundamental properties of representation geometry that shape inference ability of deep neural networks. These include signal-noise factorization (SNF), the ability to segregate signal from noise, and signal-signal factorization (SSF), the ability to segregate task-specific and task-irrelevant signals. Here, we built regularizers that reinforce these two properties during training. We compared networks trained with these regularizers to

L_2

-regularized baseline networks on the CIFAR-100 classification task to understand how our regularizers shape representation geometry and impact performance on a well-known computer vision baseline. Enhancing SNF via regularization improved model performance but enhancing SSF did not. Motivated by biomedical applications, we investigated how our regularizers affected performance on the BloodMNIST dataset treated with MedMNIST-C corruptions at five severity levels, and found even larger performance gains using the SNF regularizer. To understand the mechanism by which SNF-regularization produces improved performance, we analyzed the nuisance subspaces across regularization regimes, finding that the SNF-regularized models represent noise in distinct subspaces, separate from class-relevant signal. Because this geometry is explicit, the dominant corruption-induced directions can be estimated on held-out data and projected out of the representations. This manipulation led to a substantial gain in accuracy. These results show that regularizers that enforce signal-noise factorization can produce substantial improvements on computer vision tasks that contain out-of-distribution image distortions at inference time. They also highlight how shaping representations affects model performance: isolating nuisance variables from categorical ones is more important than maintaining factorized representations of categorical variables.

Sakin Kirti, Joel Zylberberg

Bioinformatics IDP, Jules Stein Eye Institute University of California, Los Angeles Los Angeles, CA 90095, USA · Jules Stein Eye Institute University of California, Los Angeles Los Angeles, CA, 90095, USA

Regularization Subspace Representation Space Factor

Sep 30, 2026cs.LG

Learning Functional Subspaces for Neural Network Compression

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.

Massimo Bini, Anders Christensen, Stephan Alaniz +5

Helmholtz Munich · Technical University of Munich · MCML +6

Large Language Model Compression Subspace Low-Rank Structure Data Compression Methods Peak Memory

Sep 30, 2026cs.LG

Awakening of the Buddha: Subspace Learning During Population-Loss Plateaus

Population loss can remain nearly constant while a neural network learns a substantially more predictive representation. We establish this separation for two-layer ReLU and leaky-ReLU networks trained on Gaussian inputs by simultaneous fixed-step population gradient descent on all parameters. For structured additive teachers whose links are positive mixtures of Gaussian-damped cubics in

H^1(γ)

, we give explicit conditions under which small IID Gaussian initialization yields a high-probability guarantee: at a checkpoint during a high-loss plateau, minimum alignment between the rank-

r

teacher subspace and the leading

r

-dimensional eigenspace of the predictor's average gradient outer product (AGOP) increases by at least

1/2

, and the minimum refit MSE under unchanged coefficient budgets decreases by more than

0.399

, both relative to initialization. The same trajectory subsequently attains a trained loss below every value in the plateau window. A complementary result treats unequal-weight cubic teachers and small additive Sobolev perturbations using projected-feature refits. For SwiGLU networks with an exactly fitted intercept, we prove leading-AGOP alignment during a loss plateau at fixed width and dimension as Gaussian initialization vanishes, for square-integrable teachers with nonzero Hermite content of degree one, two, or three. A rank-one cubic specialization also gives simultaneous unrestricted-refit gains at a prescribed width. An approximation lower bound further shows that certain interaction targets retain nonzero error when ridge neurons are restricted to shared orthogonal axes within the teacher subspace. Population-moment experiments with ReLU students across 21 teachers and 50 initializations per teacher complement the analysis.

Akash Kumar

Department of Computer Science & Engineering University of California-San Diego

Anisotropic Loss Landscapes Singular Learning Theory Subspace Teacher Population Dynamics Bodhi

Sep 30, 2026cs.CL

Concept Subspaces Compute Beyond the Logit Lens: A Weights-Only Test for Locating Representations Upstream of Readout

A concept subspace's effect on model behavior does not establish how it relates to the output readout. We introduce a two-sided geometric diagnostic that measures an extracted subspace's overlap with the dominant right-singular directions of the unembedding matrix, evaluated against output-oriented positive controls. Given an extracted basis, the raw diagnostic requires only model weights. Our testbed is the Format-Agnostic Reasoning Subspace (FARS), a ten-dimensional basis extracted from eighteen reasoning concepts expressed in six surface forms. Across nine rank-matched estimators and twenty-six models, four activation-derived concept estimators carry only 0.38--0.80% mean energy in the top-ten readout span. Final-layer PCA carries 3.56%, exceeding FARS in 25 of 26 models. A same-layer next-token control, evaluated using a fitted linear translator for depth matching, carries approximately thirteen times more energy than FARS, with separation in all 25 tested models. Re-extracting FARS on ten disjoint concepts yields 62--100% cross-format retrieval across twenty-four generative models, demonstrating transfer of the extraction procedure rather than a fixed basis. A complementary four-model, three-seed intervention study finds model-dependent source-directed effects that remain well below full-vector replacement. Together, the geometry and intervention controls distinguish concept structure from dominant readout directions while limiting claims of causal sufficiency.

Aojie Yuan, Zhiyuan Julian Su, Haiyue Zhang +1

University of Southern California · Duke University · University of Michigan

Subspace Readout Logit Lens

Sep 30, 2026cs.LG

QuanVI: Score-based Variational Inference via Quantum Maximally Mixed States

Score-based variational inference (VI) provides an alternative to Kullback--Leibler (KL)-based VI by minimizing the Fisher divergence between the variational distribution and the target. A prior score-VI approach formulates this optimization as an eigenvalue problem, with the variational distribution constructed from low-energy eigenstates. However, this eigenvalue-based formulation faces two high-dimensional obstacles: an intractably large parameter count due to exponential scaling and non-uniqueness of individual eigenvectors in degenerate or nearly degenerate low-energy subspaces. We propose QuanVI, a scalable quantum-inspired algorithm that combines a mixed-state density-operator formulation with a quantum tensor network (QTN) parameterization using the matrix product operator (MPO) structure. In degenerate low-energy subspaces, the density-operator formulation represents the subspace by its maximally mixed state rather than relying on a non-unique individual eigenvector, while the QTN parameterization compresses the density operator to avoid exponential parameter growth. Experiments and ablations show that QuanVI agrees with exact solutions in low dimensions and scales to high-dimensional synthetic and Bayesian posterior-approximation benchmarks, including challenging non-Gaussian targets.

Yuchen Cong, Zerui Tao, Chao Li +2

Juntendo University · RIKEN-AIP

Variational Quantum Algorithms Variational Inference Spectral Decomposition Subspace Mixtures

Sep 29, 2026cs.AI

Transolver- $σ$ : Joint Spectral-Physical Subspace Modeling for Neural PDE Solving

Neural solvers offer efficient surrogates for numerical simulation of partial differential equations (PDEs). For time-dependent problems, strong one-step accuracy does not necessarily translate into reliable autoregressive rollout. We observe that a solver based only on physical-state modeling can achieve lower one-step error, whereas its spectral-only counterpart can become more accurate at later rollout steps. Motivated by this observation, we present Transolver-

σ

, a neural PDE solver based on joint spectral--physical subspace modeling. Within each block, adaptive physical-state interactions and spectral transformations are modeled in dedicated latent subspaces, whose responses are recomposed to enable information exchange between the two representations. Within the physical subspace, we introduce Slice-Residual Physics-Attention (SRPA), which preserves an explicit slice-space identity path while retaining learnable cross-slice interaction. In parallel, an axis-factorized Fourier operator captures global spectral structure. Across five well-established PDE benchmarks spanning steady-state prediction and time-dependent dynamics, Transolver-

σ

achieves state-of-the-art with a benchmark-averaged relative error reduction of 33.4% over the strongest baseline for each metric, while consistently improving autoregressive rollout over single-operator counterparts. Transolver-

σ

further delivers strong gains on coupled multiphysics systems and real-world fluid and combustion measurements from RealPDEBench, demonstrating its effectiveness beyond standard simulation benchmarks.

Haonan Shangguan, Hang Zhou, Haixu Wu +3

School of Software, BNRist, Tsinghua University · MIT CSAIL

Neural Partial Differential Equation Solvers Subspace

Sep 28, 2026cond-mat.mtrl-sci

Measuring trainable degrees of freedom in materials graph neural networks: a random-subspace intrinsic dimension analysis

Final predictive accuracy is the standard basis for comparing graph neural networks (GNNs) in materials-property prediction, but it does not show how strongly performance depends on access to trainable parameter-space directions. Here, we introduce trainable-degree dependence as a complementary characterization of materials GNN learning. Using random-subspace intrinsic-dimension analysis, we train CGCNN, ALIGNN, and DimeNet++ in randomly oriented parameter subspaces across six prediction tasks and measure how performance recovers as independent trainable degrees of freedom are restored. The resulting recovery curves separate endpoint accuracy from the trainable-dimensional demand required to recover it. They reveal distinctions that final errors alone miss: metallic classification and log-bulk-modulus regression recover near-reference performance from small fractional subspaces, formation-energy and band-gap prediction show stronger architecture dependence, and phonon prediction is most sensitive to dimensional restriction. Dataset-size sweeps show that band-gap models require larger fractional subspaces as training data grows, whereas formation-energy and bulk-modulus responses are more stable. A width sweep shows that fractional thresholds can remain stable while absolute threshold dimensions increase with model size. Random-subspace analysis therefore provides a targeted stress test for how materials GNNs use their optimization space.

Shehroz Ahmad Shoaib, Kangming Li

Physical Science and Engineering Division, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia. · Department of Electrical Engineering, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia.

Graph Neural Networks Subspace

Sep 28, 2026cs.CV

Handwritten Text Recognition Lives in the High-Pixel Variance Subspace

In self-supervised pretraining for Handwritten Text Recognition (HTR), pixel reconstruction methods outperform contrastive methods, unlike in natural-image classification. We argue that this difference follows from where discriminative signal lies in pixel space: for HTR, it is concentrated in high-variance directions and largely absent from low-variance ones. This predicts that objectives preserving high-variance pixel content will transfer best. We test six SSL methods from three families (pixel-grounded MIM, JEPA, and contrastive) under matched encoder, data, and evaluation protocols on six handwriting benchmarks across five languages. With full labels, pixel-groundrounded SSL achieves the lowest CER on every benchmark and both frozen probes, exposes per-position character information that other families recover only through the readout, and is the only family to benefit from pretraining on real handwriting. Pixel-grounded representations are also more label efficient. Across datasets, encoder alignment with the high-variance pixel subspace predicts CER within every method. With a pretrained LLM decoder, a frozen pixel-grounded encoder is competitive with fully fine-tuned supervised baselines; full fine-tuning achieves the lowest mean CER and ranks first or second on every benchmark. These results show that the value of pixel reconstruction depends on where discriminative signal lies in the input.

Carlos Garrido-Munoz, Jorge Calvo-Zaragoza

University of Alicante, Spain

Handwritten Text Recognition Handwriting Self-Supervised Learning Character Error Rate Frozen Encoder Text Analysis and Detection+2

Sep 28, 2026cs.LG

AIM-ZO: Activation-Informed Subspace Maintenance for Zeroth-Order LLM Fine-Tuning

Zeroth-order (ZO) optimization offers a memory-efficient alternative for LLM fine-tuning by estimating updates only from forward evaluations of perturbed parameters, without backpropagation or activation storage. However, in billion-parameter LLMs, isotropic perturbations often waste many forward evaluations on weakly informative directions. To make these evaluations more informative, existing ZO methods restrict perturbations to low-dimensional subspaces. Yet the quality of these subspaces is critical: overly compressed or poorly maintained spaces can miss useful update directions. To obtain a high-quality subspace for ZO updates, this paper proposes AIM-ZO, a ZO fine-tuning method based on Activation-Informed Subspace Maintenance. AIM-ZO uses forward activations as local directional information and continuously integrates them into a broad, evolving subspace over training. To access broader gradient-relevant structure while keeping individual perturbations low-dimensional, AIM-ZO activates only a smaller set of shared and sampled directions, decoupling the maintained width from the active width. We evaluate AIM-ZO across 5 LLMs and 11 downstream tasks under matched forward-evaluation budgets; its six-task average exceeds the strongest fully evaluated ZO baseline by 1.26 percentage points on OPT-2.7B and MeZO by 2.85 percentage points on OPT-30B. Our code is available at https://github.com/EkkoXy/AIM-ZO

Yue Xie, Zhi Zheng, Yunpeng Ba +5

Southern University of Science and Technology · National University of Singapore · Huawei Technologies Ltd. +1

Large Language Model Fine-Tuning Continual Fine-Tuning Model Fine-Tuning Zeroth-Order Subspace

Sep 28, 2026cs.LG

PMOPD: Task Ordering, Cycling, and Parameter-Update Subspace Protection in Multi-Teacher On-Policy Distillation

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

Youzhi Liu, Ruobing Zheng, Boyuan Tong +5

Ant Group

Aware Heterogeneous Multi-Teacher Multimodal On-Policy Distillation Online-Policy Distillation Subspace Order Matters

Sep 28, 2026cs.CV

SAGE: Subspace Alignment for Classifier-Free Guidance in Mixture-of-Experts Diffusion Models

Diffusion Transformers with Mixture-of-Experts (MoE) routing are a leading recipe for scaling generative models. Classifier-Free Guidance (CFG) is essential for generation quality, yet excessively high guidance scales trigger collapse. We identify a previously unreported failure mode in their combination: the two CFG branches route independently, so their realized activations occupy different subspaces. The unconditional write then leaves the conditional subspace, and CFG amplifies that residual linearly in the guidance scale. We propose SAGE, a training-time regularizer that aligns unconditional MoE activations to the conditional subspace without restricting routing diversity, at zero inference cost. Toy experiments show that SAGE dramatically suppresses extreme drift by 9.2x. When scaled to a 1B-parameter text-to-image model, SAGE significantly improves generation quality, delivering a 9.3% boost in peak DPG-Bench performance. Extensive experiments demonstrate that SAGE consistently outperforms the baseline.

Boyu Zhang, Yangming Cheng, Ning Zhang +5

Alibaba Token Hub, Alibaba Group

Classifier-Free Guidance Generative Models Mixture-Of-Experts Subspace

Sep 28, 2026cs.LG

Distribution-Conditioned Task Routing for Class-Incremental Learning

Parameter-efficient adaptation enables continual learners to acquire task-specific knowledge through compact model updates while maintaining strong within-task performance. However, class-incremental inference requires each input to be classified among all classes seen so far without access to its task identity. For learners equipped with task-specific parameter-efficient modules, this introduces a critical task-routing challenge beyond catastrophic forgetting. We study post-hoc task routing without retraining the learner or introducing a separately trained router. Such training-free inference-time calibration remains comparatively underexplored in parameter-efficient class-incremental learning. We identify three sources of routing error (feature-level, task-level, and class-level misalignment) and propose Feature Distribution Calibration (FDC). Its three components address these misalignments: Task Subspace Filtering (TSF) suppresses feature components outside each task's principal subspace, Residual Likelihood Calibration (RLC) evaluates the typicality of its subspace residual, and Prototype Affinity Calibration (PAC) measures compatibility with the task's class prototypes. Experiments demonstrate plug-and-play applicability to eight parameter-efficient class-incremental methods using a shared encoder. With one component configuration selected per method across all five benchmarks, FDC improves final accuracy in all 40 method-dataset pairs by 4.39 percentage points on average. Enabling all components improves 35 of the 40 pairs, with an average gain of 4.45 points. When applied to a simple baseline, FDC achieves strong overall performance.

Longhuan Xu, Zhipeng Zhou, Wei Ji +3

Nanjing University · Nanyang Technological University · Shanghai Jiao Tong University

Class-Incremental Learning Subspace

Sep 28, 2026cs.LG

SPACE-LoRA: Allocating Activation-Subspace Protection for Continual Learning

This study addresses the catastrophic forgetting problem that occurs when sequentially learning successive tasks using Low-Rank Adaptation (LoRA) from a lifelong learning perspective. While existing approaches have primarily constrained parameter updates or learning subspaces to reduce interference with past knowledge, they have not fully considered additive interference. This occurs when a newly added residual adapter on top of a fixed past model generates non-zero responses along input directions important for old tasks, thereby altering previous predictions. To this end, we propose Subspace Protection with Allocated Capacity for Efficient Continual Adaptation (SPACE-LoRA). SPACE-LoRA directly suppresses the responses of the new residual branch along input activation directions that are important for old tasks and adaptively determines the protection coverage for each module based on past-task sensitivity estimated via a common Fisher sensitivity-based coverage target. Under a fixed LoRA rank, this approach adaptively adjusts module-specific protection coverage while suppressing interference along input directions sensitive to old tasks. We assess the effectiveness of activation-subspace protection in mitigating catastrophic forgetting and examine the role of sensitivity-guided protection in continual learning across diverse tasks. Code is available at https://anonymous.4open.science/r/SPACE-LoRA-7864.

Seunghyun Yoo, Kiseok Kim, Hyeontae Joo +2

School of Electrical Engineering Korea University Seoul, Republic of Korea

Replay-Based Continual Learning Catastrophic Forgetting Continual Learning Parameter-Efficient Adaptation Subspace

Sep 28, 2026cs.LG

FAST-Brain: A Flow-Aligned Spatio-Temporal Surrogate Brain Model

Modeling resting-state functional magnetic resonance imaging (rs-fMRI) data is crucial for understanding brain-wide neural activity. However, traditional methods struggle to capture complex temporal dynamics over long horizons, to account for the brain's anatomical spatial structure, and to model high-dimensional ambient signals that lie on a low-dimensional intrinsic subspace. We propose FAST-Brain, a unified flow-aligned spatio-temporal surrogate brain model that addresses all three challenges. At its core is a flow-aligned generative framework that directly predicts the clean blood-oxygen-level-dependent (BOLD) signal, paired with a graph convolutional network that captures spatial structural constraints and a Transformer that models long-range temporal dependencies. Theoretically, we show that under a low-dimensional subspace assumption, the approximation error of our model scales with the intrinsic dimension rather than the ambient dimension, which justifies our direct modeling of the BOLD signal. Extensive experiments on synthetic and Human Connectome Project datasets demonstrate that FAST-Brain achieves state-of-the-art performance in recovering functional connectivity, effective connectivity, and the implicit low-dimensional signal subspace.

Shucheng Liu, Changchun Shi, Kai Zhang +1

University of North Carolina at Chapel Hill · London School of Economics and Political Science

Subspace Spatiotemporal Sequence Modeling

Sep 28, 2026cs.LG

Learning to Steer, Steering to See: Unveiling the Geometry of RLVR in Large Language Models via Trainable Vectors

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

Yuchen Cai, Ding Cao, Qixiang Yin +10

USTC · Tencent Hunyuan · BUPT

Reinforcement Learning With Verifiable Reward Large Language Model Training Steering Subspace

Sep 27, 2026cs.RO

Principal Steering Subspaces for Online Adaptation of Frozen Generative Robot Policies

Generative robot policies provide expressive behavior priors, but updating a large diffusion or flow-matching model through online interaction is costly. Latent-space reinforcement learning avoids updating the pretrained generator by controlling its initial sampling noise, yet high-dimensional noise can have strongly anisotropic effects on decoded actions. We introduce Principal Steering Subspaces (PSS), a forward-query interface that constructs a fixed low-dimensional control basis from finite-difference decoder responses. Soft Actor-Critic controls the leading response directions, while the orthogonal complement is independently resampled from the Gaussian prior at each query. On three RoboMimic tasks with diffusion and flow-matching policies, response spectra reveal substantial concentration. Across five matched task-generator pairs, the training curves indicate that PSS generally converges faster and exhibits more stable late-training behavior than full-latent control, while achieving stronger final performance overall. Controlled Diffusion-Square ablations further show that leading-response directions outperform random and least-responsive subspaces of equal dimension. We further integrate PSS with a frozen, closed-source 3B-parameter vision-language-action (VLA) policy in a humanoid learning system with synchronous transition collection, reset-time optimization, and latency-aware asynchronous deployment. In an exploratory screwdriver-placement evaluation, success is observed in 2/10 trials for the frozen VLA policy and 6/10 after SAC+PSS adaptation. These results support decoder-response geometry as a practical basis for online adaptation of frozen generative robot policies.

Jialeng Ni, Nathan Zhao, Kunpeng Song

University of Michigan, Ann Arbor, MI, USA. · XPENG Robotics, Santa Clara, CA, USA.

Robot Policies Latent Action Models Subspace Prior Knowledge Integration

Sep 27, 2026cs.CV

PulseQuant: Propagation-Guided Subspace Correction for 4-Bit Video Diffusion Transformers

Quantization errors in video diffusion transformers can be amplified or attenuated by subsequent denoising updates, making local reconstruction error an incomplete predictor of final impact. We introduce PulseQuant, a 4-bit post-training quantization method that combines trajectory sensitivity with activation geometry to guide offline calibration. Isolated block--step interventions estimate propagation risk, which prioritizes sensitive trajectory states during row-radius selection. With these radii fixed, response-subspace correction uses neighboring-code edits to reduce residual components along dominant activation directions. Both stages preserve the original 4-bit weight representation. Controlled interventions show that short-horizon propagated error predicts final latent error more reliably than immediate block-output error, supporting calibration beyond local reconstruction objectives. Evaluations on Wan models, Self Forcing, and MiniMax-H3 demonstrate improvements in key consistency and dense-reference metrics while remaining competitive on other attributes across model scales and generation paradigms.

Yutong Wang, Xingtong Ge, Enhuai Liu +4

The University of Sydney · The Hong Kong University of Science and Technology · The Chinese University of Hong Kong +1

Sep 27, 2026cs.LG

Towards Identifiable Representations under Misspecified Structure

The presence of noise that depends on the latent variables poses a fundamental challenge to identifiability. Existing results rely on conditional independence among the observations given the latent variables. We study a more general \emph{misspecified structure}, where this conditional factorization does not hold, and establish both precise and approximate identifiability guarantees. We characterize structural misspecification as a perturbed factor analysis problem. For precise identifiability, we establish subspace identifiability under spectral separation and controlled perturbation, followed by component-wise identifiability under structural sparsity. When the precise condition is not guaranteed, we derive an approximate subspace-identifiability theorem. Based on these results, we develop an unsupervised variational estimator for recovering latent variables. Experiments demonstrate the effectiveness of the proposed framework.

Yuke Li, Yujia Zheng, Ziyi Chen +2

University of Maryland College Park, College Park, MD, USA · Carnegie Mellon University, Pittsburgh PA, USA

Identifiability Latent Variable Exploratory Factor Analysis Unsupervised Subspace Sparsity+1

Sep 27, 2026cs.RO

Q-WAM: 4-Bit Quantization of World Action Models with Action-Subspace Protection

World Action Models (WAMs) jointly generate video and robot actions through iterative diffusion and perform strongly in robotic manipulation. However, their prohibitive compute and memory costs pose substantial deployment challenges. Post-training quantization (PTQ) can reduce these costs, but existing PTQ methods such as smoothing and rotation are insufficient to maintain the precision of action generation. To overcome this limitation, we propose Q-WAM, a new 4-bit weight-activation quantization for WAMs that preserves the actions the model generates. Specifically, we introduce the \textit{Action Observability Gramian (AOG)}, which measures how much rounding errors in each weighted combination of a layer's input channels change the final action through all denoising steps. We also develop Action-Subspace Protection (ASP), which keeps the few most action-sensitive channel combinations in a tiny 16-bit low-rank branch and quantizes the complementary weights and activations to 4 bits, both as dense matrix multiplications that run efficiently on GPUs. Finally, to preserve action quality with minimal overhead, we identify the experts that matter most for the generated action by aggregating the AOG-derived action mass across the layers of each expert and apply ASP only to those experts. We evaluate Q-WAM on three WAMs, both in simulation and in real-world deployment. On the RoboTwin 2.0 benchmark, it reaches 89.6--93.0% average success rate, within 1.1 percentage points of the 16-bit models, while reducing the memory of the quantized blocks by 3.1--3.4

\times

. Our method outperforms the strongest baseline, SVDQuant, by 2.5--8.7 percentage points. On a Unitree G1 humanoid and a bimanual UR3 robot, it improves success over SVDQuant by 12.8-17.6 percentage points.

Arash Akbari, Arman Akbari, Jingwu Luo +7

Northeastern University · EmbodyX · University of Minnesota - Twin Cities +1

Faster-Wam World Models Subspace Channel

Sep 24, 2026cs.CG

It's the Geometry, Not the Model: Effective Rank and Subspace Alignment in Functional Connectivity Classification

Resting-state functional connectivity (FC) is widely used to classify brain phenotypes and disorders. Most pipelines use the full connectome and seek gains through model design. We instead examine how FC geometry constrains classification and cross-site transfer. Across-subject FC variation concentrates in a small effective subspace, suggesting substantial redundancy in nominal dimensions. Across cohorts, these subspaces may differ in orientation even when their effective ranks are comparable, potentially limiting transfer. Across 2,330 subjects from HCP, ABIDE, and ADHD-200, effective-rank analysis reveals strong spectral concentration. Projection onto leading components at the effective-rank scale recovers most of the full-FC classification performance. In ABIDE, site-specific effective subspaces are weakly aligned, and their principal-angle overlap predicts pairwise transfer after covariate adjustment despite comparable per-site effective ranks. Controlled rotations that alter subspace orientation while preserving the mean and covariance spectrum drive transfer toward chance, whereas displacement-matched label-orthogonal rotations do not. These results identify subspace orientation as a key factor in transfer degradation under controlled perturbations. This study offers a geometric diagnostic of FC generalization and suggests evaluating cross-site harmonization by its ability to align effective subspaces alongside classification accuracy.

Xiao Fan, Jingyuan Li, Yubo Han +6

Functional Connectivity Human Connectome Project Subspace Controlled Comparison Ranking Covariance

Sep 23, 2026cs.CV

Looks the Same, Answers Differently: Flip-Direction Steering for Robust Vision-Language Reasoning

Vision-language models (VLMs) achieve strong visual reasoning performance, yet subtle changes from routine image capture and processing can alter their reasoning trajectories even when images appear nearly identical. In long-horizon generation, the resulting activation shifts may accumulate across decoding steps, progressively altering reasoning tokens and ultimately changing the final answer, a phenomenon referred to as answer flips. To address this instability, we propose FlipDir (Flip-Direction Steering), a training-free inference-time method that estimates a low-rank flip-inducing activation subspace from contrastive pairs of original and answer-flipping inputs and selectively steers hidden states during decoding. A margin-based gate limits subspace attenuation to uncertain decoding steps, recovering original predictions while preserving stable ones. To evaluate robustness beyond accuracy or consistency on fixed test sets, we introduce VisFlip, a benchmark framework that constructs evaluation groups for a target model and visual variation setting to separately assess recovery of original predictions and preservation of stable ones. VisFlip spans nine dataset-variation combinations across scientific reasoning, robot-scene understanding, and medical VQA, covering subtle visual variations common in each domain. Experiments across 18 settings demonstrate that FlipDir consistently outperforms existing methods on the combined recovery and preservation metric. We will make our code publicly available.

Yeonsung Jung, Joonhyun Jeong, Hoang Pham +4

Graduate School of AI, KAIST · NAVER Cloud · Department of Computer Science and Engineering, The Ohio State University +3

Visual Reasoning Steering Answer Subspace

Sep 23, 2026cs.CL

A Native-Reference Coordinate Geometry for L2 Pronunciation Deviation Using Self-Supervised Speech Models

Self-supervised speech models encode rich phonetic information, but it remains unclear how to transform this information into interpretable metrics for second-language (L2) pronunciation assessment in spontaneous speech. We propose a native-reference coordinate geometry in which phone-class averages from native speech define a low-dimensional reference subspace, and L2 speech is evaluated by its distance to matching native phone-class coordinates. Unlike prior distance-based approaches, our method does not require parallel recordings with matched linguistic content or dedicated pronunciation labels. Across different self-supervised encoders and modeling choices, the resulting native-reference distances show negative Spearman correlations up to -0.5 with speaking proficiency, indicating that higher-proficiency speakers tend to lie closer to the native-reference space.

Tina Raissi, Nhan Phan, Mikko Kurimo

Department of Information and Communications Engineering, Aalto University, Espoo, Finland

Pronunciation Assessment Grapheme-To-Phoneme L2 Subspace

Sep 22, 2026stat.ML

SuperPCA: subspace analysis and an efficient algorithm for high-dimensional PCA

Principal component analysis (PCA) is a fundamental tool to reduce the dimensionality of the data in many applications. PCA finds a few signal directions that contain most of the variability of the data by computing the eigenvectors of the sample covariance matrix. In this work, we focus on the spiked covariance model, in which the data vectors are defined by a few orthogonal signals plus an isotropic Gaussian noise, and our goal is to estimate one or more of the leading signals. Our main theoretical finding is that the subspace spanned by several leading eigenvectors of the sample covariance matrix contains significant information about the desired signals long before the individual eigenvectors converge to the population principal components. To prove this, we derive a posteriori bounds for the angle between the subspace spanned by the desired population signals and the subspace obtained from the sample using perturbation theory for singular vectors. This leads to a new algorithm, SuperPCA (SUbsPace subsamplER PCA), which capitalizes on an approximate eigenspace of the sample covariance matrix to find the leading signals far more efficiently and accurately than classical PCA in the high-dimensional, multi-signal setting. SuperPCA exploits only a small number of subsampled coordinates of the data, which can lead to tremendous savings in data acquisition cost, especially when the signals are approximately sparse. For the same number of measurements, SuperPCA can offer a factor

10

improvement in accuracy compared to the classical PCA method.

Irina-Beatrice Haas, Maike Meier, Yuji Nakatsukasa +1

Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK · Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen, 9700 AB, Groningen, The Netherlands · Institute of Mathematics, EPFL, 1015 Lausanne, Switzerland

Principal Component Analysis Subspace Covariance Spectral Decomposition

Sep 21, 2026cs.CV

Beyond Uniform Subspaces: Spectrum-Aware and Depth-Adaptive Fusion for Multi-Task Model Merging

Model merging aims to consolidate multiple task-specific models without access to extra training process. However, existing subspace-based methods largely rely on a uniform treatment of task updates, overlooking their intrinsic spectral and depth-wise heterogeneity. We identify two key deviations from this assumption: different tasks require different subspace capacity and exhibit different tolerance to spectral transformation, while subspace projection introduces depth-dependent distortion. Based on these observations, we propose SADA-Merging, a spectrum-aware and depth-adaptive framework for data-free model merging. SADA-Merging allocates task-specific subspace capacity according to spectral complexity, adapts spectral preservation according to task-wise plasticity, and applies depth-dependent anchoring to compensate for projection-induced distortion. This enables the fusion process to adapt to both the intrinsic geometry of each task and its sensitivity across network depth. SADA-Merging operates directly on task updates and is applicable to both full fine-tuning and LoRA settings. Extensive experiments demonstrate consistent improvements over existing data-free merging methods across different task scales and adaptation settings.

Ruxi Gu, Zilei Wang, Wei Wang

Department of Automation, University of Science and Technology of China · State Key Laboratory of General Artificial Intelligence, BIGAI · School of Computer and Communication Engineering, University of Science and Technology Beijing

Continual Model Merging Multi--Task Learning Subspace Fourier

Sep 20, 2026cs.CV

CE $^4$ L: Continual Ego, Exo, and Ego-Exo Learning

Perception for embodied agents is video-based, often multi-view (ego, exo, or both), and inherently continual, with simultaneous task and viewpoint shifts. Yet continual learning (CL) remains dominated by exo-only recognition tasks, obscuring behavior under these real-world coupled shifts. We introduce Continual Ego, E}xo, and Ego-Exo Learning (CE

^4

L), a unified multi-view CL benchmark spanning four representative tasks: cross-view referenced skill assessment, temporal action segmentation, cross-view association, and action anticipation & planning. CE

^4

L highlights challenges largely absent in prior CL benchmarks, including cross-view correspondence, view-dependent asynchrony, and heterogeneous semantic objectives. To this end, we propose Video Incremental Subspace-routed Task Adapters (VISTA), a parameter-efficient baseline method that stores task-specific updates in lightweight adapters and performs training-free routing via residual distance to task-specific whitened subspaces estimated from second-order statistics. Extensive experiments demonstrate the significantly varied efficacy of representative CL methods across CE

^4

L settings, while VISTA is consistently competitive and achieves state-of-the-art overall performance. Our source code for benchmarks and methods is available at https://github.com/AnAppleCore/CE4L .

Hongwei Yan, Kanglei Zhou, Yuchen Liu +3

School of Life Sciences, IDG/McGovern Institute for Brain Research, Tsinghua University, Beijing, China · Tsinghua-Peking Center for Life Sciences · Department of Psychological and Cognitive Sciences, Tsinghua University, Beijing, China +2

Replay-Based Continual Learning Continual Learning Multi--Task Learning Embodied Agents Subspace

Sep 17, 2026stat.ML

Online Supervised Dimension Reduction with Random Features: Diagnostics and Computational Trade-offs

Accurate optimization of a supervised spectral objective need not produce an accurate population subspace or a better predictive representation. We investigate these distinctions for Online Kernel Supervised Principal Component Analysis (OKSPCA), which combines a centered cross-moment in finite random-feature coordinates with an Adam-style orthonormal basis update for an established objective. Fixed-map consistency, concentration and perturbation results describe the estimator and its exact subspace; same-target comparisons then assess the practical iterate separately. Across six predictive benchmarks, performance depends on the declared pipeline: replacing the tracker with the exact empirical target leaves the two regression deficits largely unchanged. Direct classification-rank models capture nearly all terminal objective energy on average, but a saved intermediate state exhibits substantial geometric deviation; a controlled sample-size study further separates empirical accuracy from population recovery. In distinct numerical-service workloads, exact on-request computation is faster in the tested classification settings, whereas Adam saves time relative to the tested full thin-SVD service for some dense wider-regression requests, alongside persistent geometric error. These diagnostics limit explanations based solely on terminal optimization accuracy and distinguish numerical cost from quality, rank coverage and freshness; they establish neither practical-tracker convergence nor predictive or deployment benefits from basis availability.

Zhenlin Yao, Wei Xiong

School of Statistics University of International Business and Economics Beijing, China

Principal Component Analysis Subspace Dimensionality Reduction Adam Random Diagnosis

Sep 17, 2026cs.LG

Compressed Active Subspaces for Scalable Bayesian Inference

Active subspace methods provide a framework for quantifying predictive uncertainty in high-dimensional models by identifying and performing inference along parameter directions that have the greatest influence on the model output. However, the construction of active subspaces requires storing many full-dimensional model gradients, which becomes prohibitive as model size increases. We address this limitation by proposing Compressed Active Subspaces (CAS), a scalable approach that first maps the model parameters to a compressed space using a structured isometric embedding and then constructs the active subspace within this reduced parameterization. Our approach substantially reduces the memory required for active subspace construction and enables Bayesian inference for large models where standard active subspace methods become impractical. We demonstrate the scalability of CAS on neural networks of increasing size while maintaining predictive performance and robust uncertainty estimates.

Thomas Flynn, Sanket Jantre, Byung-Jun Yoon +1

Computing and Data Sciences Directorate, Brookhaven National Laboratory, Upton, NY, USA · Department of Electrical & Computer Engineering, Texas A&M University, College Station, TX, USA · Mathematics and Computer Science Division, Argonne National Laboratory, Lemont, IL, USA

Subspace Bayesian Inference Predictive Uncertainty

Sep 16, 2026eess.SP

Learning Array Signal Topologies as Conditional Neural Manifolds

Subspace methods such as multiple signal classification (MUSIC) achieve super-resolution direction of arrival (DoA) estimation by exploiting the orthogonality between the array manifold and the noise subspace of the measurements. Their accuracy therefore depends on the assumed manifold and degrades under model mismatch, while parameters not identifiable from the spatial manifold cannot be recovered. In this work, we propose the conditional neural manifold (CNM), which replaces the fixed manifold with an observation-conditioned mapping from source parameters to steering vectors. An encoder maps the snapshots to a latent scene representation that conditions a zero-initialized neural field over the parameter space. The manifold is learned without steering-vector supervision by shaping the resulting MUSIC landscape. Since the correction acts on the manifold rather than on the estimator, it can be used by other manifold-based methods without modification. The CNM restores resolution under array imperfections, colored noise, correlated sources, and near-field propagation, and resolves the angle-frequency ambiguity inherent to the nominal spatial manifold.

Julian P. Merkofer, Vincent van de Schaft, Ruud J. G. van Sloun

Eindhoven University of Technology, Eindhoven, The Netherlands · University Medical Center Utrecht, Utrecht, The Netherlands

Microphone Array Subspace Manifolds Generative Super-Resolution

Sep 16, 2026cs.LG

Label-free steering: Compressing test-time reinforcement learning into bias-only subspaces

Test-time reinforcement learning (TTRL) enables models to improve their reasoning without relying on labeled training data, but existing approaches typically optimize a large fraction of the model parameters. This raises a natural question: can effective test-time adaptation emerge when both the reward signal and the optimization space are severely restricted? We answer this question with label-free bias-only TTRL, which uses majority-vote pseudolabels as rewards and optimizes only ~100K bias parameters while keeping the pretrained backbone frozen. On MATH-500, our approach reaches 76.67% accuracy with Qwen2.5-7B, slightly exceeding our own labeled bias-steering reproduction while optimizing 76,000x fewer parameters than full-parameter TTRL. The same training procedure improves performance across vision-language and audio reasoning tasks, including MathVista, AI2D, LogicVista, and MMAU. We further show that the learned steering vectors transfer to 4,500 held-out MATH problems, indicating that the adaptation is not limited to the problems used during test-time optimization. Finally, we analyze why this highly restricted adaptation can work, showing that majority-vote reliability improves with rollout consensus and that bias subspaces with greater accessible gradient energy exhibit stronger downstream trainability. These results demonstrate that substantial test-time adaptation can emerge from optimizing a tiny bias-only subspace using entirely label-free rewards.

Naveen Vakada, Mingyuan Li, Shaoxiong Ji

University of Turku, Finland · ELLIS Institute Finland

Stable Test-Time Adaptation Test-Time Adaptation Subspace Steering Vista

Sep 16, 2026math.OC

Gradient Descent with Stochastic Subspaces via Persistence of Memory

Stochastic subspace methods have gained popularity as gradient descent based techniques for large scale optimisation problems, especially in distributed settings. In this paper, we introduce the technique of "persistence of memory" to greatly extend and improve the random subspace methods. To this end, we leverage a vector that is only weakly correlated with the gradient in order to provide a guiding structure to the generative process of the random subspace along which the descent is going to take place. This guidance vector may be fixed for a large number of iterations, only to be refreshed at wide intervals (on whose size we can provide guarantees in terms of problem parameters). In important machine learning settings, such as optimisation problems embodying sparsity or a minibatch structure, we show that the guidance vector can be obtained in an effective and computationally inexpensive manner by leveraging the structured properties of the problem. En route, we establish to our knowledge the first theoretical analysis of classical SSD methods for sparse functions. In a local neighbourhood of the optimum, we demonstrate an alignment phenomenon of our gradient estimates with a low-lying eigenvector of the Hessian, allowing a once-for-all computation of the guidance vector which renders the method computationally favourable even in scenarios with unstructured objectives.

Subhroshekhar Ghosh, Clement Z. Q. Ng, Pierre-Louis Poirion +1

Department of Mathematics, National University of Singapore, Singapore · Center for Advanced Intelligence Project, RIKEN, Tokyo, Japan · Department of Mathematical Informatics, The University of Tokyo, Tokyo, Japan

Stochastic Gradient Descent Gradient Descent Subspace Gradient Persistence

Sep 15, 2026cs.LG

NObSP: Functional Decomposition of Neural Networks via Oblique Subspace Projections

Understanding how deep neural networks make decisions remains a fundamental challenge. We present NObSP (Nonlinear Oblique Subspace Projections), a framework that decomposes predictions into explicit per feature contribution functions and an interaction residual. NObSP exploits the linear final layer of a trained network and uses oblique projections in sample space to reduce double counting when learned feature subspaces overlap, thereby supporting both local explanations and global functional analysis. We establish connections to functional ANOVA and the Kolmogorov-Arnold representation theorem and derive an efficient partial regression algorithm for out of sample evaluation. For convolutional networks, NObSP-CAM produces class activation maps without backward passes after a one time calibration. Experiments on tabular and vision benchmarks show faithfulness comparable to established attribution methods. On a synthetic benchmark with known component functions, NObSP obtains a Function Reproduction Score of 0.989, compared with 0.966 for KernelSHAP and 0.922 for Integrated Gradients. On TinyImageNet, contribution vector embeddings improve mean nearest neighbor class purity from 0.654 for raw activations to 0.713 and reduce mean neighbor distance by more than half. These results indicate that NObSP complements scalar attribution methods by recovering functional contribution profiles with separable positive and negative evidence.

Alexander Caicedo, Víctor De La Hoz, Santiago Alférez

Department of Electronic Engineering, Pontificia Universidad Javeriana, Bogot´a, Colombia · Pontificia Universidad Javeriana, Bogot´a, Colombia · Department of Mathematics, Universitat Polit`ecnica de Catalunya, Barcelona, Spain

Neural Network Subspace Convolutional Neural Networks Decomposition Camera

Sep 14, 2026cs.LG

Sharp Rates and a One-Line Correction for Spectral Representation Learning

A self-supervised encoder is trained once, frozen, and reused through lightweight probes on tasks nobody named at training time; the practitioner's question is when the off-the-shelf features are good enough and when they need fixing. Canonical correlation analysis, HGR maximal correlation, and the population optimum of the spectral contrastive loss all return the top-

k

singular subspace of a cross-view dependence operator, justified by isotropy: if the task prior has no directional preference, that subspace is universally optimal. We show isotropy is the wrong hypothesis. The prior enters the transfer risk only through the task covariance

Λ=\mathbb{E}[ΔΔ^\top]

, and only through its compression onto the operator's leading singular directions; what matters is not whether

Λ

is isotropic but whether its preferred directions are ordered consistently with the operator's spectrum. We prove matching two-sided rates---worst-case regret is exactly

1-1/κ(Λ)

, refines to

1-A_k

for an alignment coefficient

A_k

, localizes to the top-

2k

subspace, becomes second order under a spectral gap, and is improvable by no task-agnostic representation---and show why alignment is generic: incoherent preferences cancel in high dimension, and

T

diverse tasks force

α=\widetilde O(\sqrt{d_x/T})

, a quantitative account of why task diversity, not symmetry, makes self-supervised features transfer. The governing statistics cost

O(kd_x^2)

, and when they signal misalignment a one-line reweighting of the positive-pair term provably restores exact optimality. The result is a diagnostic that answers the practitioner's question from a small labelled budget and refuses when the task bank cannot support the width requested; on controlled data it takes a regret of

0.86

down to

0.003

, and on a CIFAR-100 encoder it correctly predicts that no correction is needed.

Dier Tang, Jing Yee Tan, Guangyue Han

Department of Mathematics, The University of Hong Kong

Self-Supervised Representations Subspace Spurious Correlations Correction

Sep 14, 2026cs.LG

Rotation-Based Subspace Tracking for Robust Kernel PCA on Streaming Data

Machine learning models process large amounts of data, and Principal Component Analysis (PCA) is a widely used technique to reduce the dimensionality of the data and extract useful features. In practice, datasets often change over time (data drift) and/or arrive one sample at a time (streaming data), making it infeasible to process the entire dataset at once in batch mode. Real-world data also often contains nonlinear patterns, which traditional PCA cannot extract. Kernel PCA addresses this by implicitly mapping samples into a Reproducing Kernel Hilbert Space (RKHS). Raw data also often contains outliers, which can have an outsized effect on the estimated subspace unless the algorithm is made robust. However, existing online robust kernel PCA algorithms are designed to converge to a subspace that is assumed to be fixed, and gradient-descent-based updates lose their effectiveness at tracking further changes once this initial alignment is achieved. This paper introduces a rotation-based update mechanism, which updates the subspace estimate by rotating it toward each new incoming feature vector in Reproducing Kernel Hilbert Space, rather than relying on gradient descent alone. We present two complementary rotation strategies, and show that the extent of rotation can be moderated by a robust influence function to mitigate the effect of outliers. Through experiments on synthetic streaming data with a known ground-truth subspace, we show that per-sample rotations converge faster than gradient descent alone, demonstrating an effective mechanism for dynamically tracking a nonlinear subspace in streaming data.

Kris Lokere, John Fossaceca

Harvard University, Cambridge, MA, USA · George Washington University, Washington, DC, USA

Principal Component Analysis Subspace Streaming Synthetic Data

Sep 14, 2026cs.LG

Single-condition neural solvers encode transferable response spaces for parametric differential equations

Operator learning for parametric partial differential equations (PDEs) typically builds global models over prescribed domains, requiring cross-condition data or costly physics-constrained training. Here we show that the output Jacobian of a neural solution model trained at one condition defines a reusable response space for cross-condition solution variations. We introduce Linearized Subspace Transfer (LST) to exploit this space and recover target solutions by minimizing the target PDE-system residual over response-space coordinates. Because any single response space has finite coverage, Active Transfer Modeling (ATM) uses post-transfer residuals as coverage indicators to selectively acquire response spaces from additional single-condition models. Across six systems, single-condition response spaces supported cross-condition transfer, with enrichment improving accuracy when added spaces expanded representation capacity. Relative to evaluated physics-informed operator baselines, ATM reduced error and offline construction cost, with orders-of-magnitude accuracy gains in representative cases and millisecond-to-second target adaptation. These results establish neural solvers as reusable local parametric models.

Wenbo Cao, Weiwei Zhang

Institute of AI for Industries, Chinese Academy of Sciences, Nanjing 211135, China · Institute of Computing Technology, Chinese Academy of Sciences, Beijing 100190, China · School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, China +2

Neural Partial Differential Equation Solvers Neural Solvers Partial Differential Equations Subspace Uncoupled Learning Responses

Sep 14, 2026cs.LG

SCOPE-OPSD: Fisher-Conditioned Privileged Subspaces for On-Policy Self-Distillation

On-policy self-distillation (OPSD) scores student-generated prefixes with a solution-conditioned self-teacher, yet transfers supervision only through next-token probabilities. We ask whether the aligned final-layer discrepancy offers a useful second channel, and how to test that channel without confusing its geometry with auxiliary strength. SCOPE-OPSD projects the privileged teacher-student residual onto a frozen rank-64 factor estimated from residual covariance and language-model-head Fisher sensitivity. It reuses the forwards already required by OPSD and adds neither rollouts nor inference-time modules. A matched Random control preserves the structured factor's rank and nonzero spectrum and uses per-arm gradient-RMS calibration, isolating the effect of the data-dependent orientation. Across the complete 25/50/75/100-step trajectories for Qwen3-1.7B, 4B, and 8B, Structured is never below Pure OPSD, with strict gains in 11 of the 12 model-checkpoint combinations and an exact tie at 4B step 25. Structured also exceeds matched Random in 10 of the 12 combinations. At step 75 on Qwen3-1.7B, Structured exceeds matched Random by 1.39 Macro Avg@12 points in each of two independent training reruns. A cross-fitted diagnostic also shows 4.40 times greater held-out privileged-gap capture than the matched random orientation. The results support a compact, Fisher-conditioned privileged subspace for short-budget OPSD.

Yunmeng Chen (Chongqing Ant Consumer Finance Co., Ltd), Kunyu Wang (Alibaba Cloud Computing Co. +18

Chongqing Ant Consumer Finance Co., Ltd. · Alibaba Cloud Computing Co., Ltd. · Chongqing University of Posts and Telecommunications

Unsupervised On-Policy Self-Distillation Rp-Opsd Subspace

Sep 13, 2026cs.LG

Refusal Reads Only a Slice of What the Model Knows: Harm-Keyed Routing and Its Exceptions Across Model Families

Alignment applied after pretraining is shallow in a measurable way: a single direction in a model's residual stream can be edited out, and the model stops refusing harmful requests. That fact says how easily refusal can be removed, not what the refusal decision was reading in the first place. We ask what it reads, and we separate that from what the model comprehends. Across four open-weight models spanning three families, moral comprehension is native to pretraining: a low-rank moral subspace crystallizes during pretraining, and alignment rotates it once without rebuilding it. The refusal gate, in contrast, is a fresh post-training construction with only a weak pretraining precursor, written into a narrow control-token channel where the refusal decision is orthogonal to the moral-judgment decision. The central result is causal and comes from one model, OLMo-3. A nested interchange rank sweep patches successively larger slices of the moral subspace between matched requests and reads how much of refusal's response transfers: as the basis widens, moral judgment keeps reading more of it, while refusal levels off at the level of a single harm direction, and about three-quarters of refusal's causal input lies outside the moral subspace altogether. Refusal reads the harm percept, not the moral content that judgment reads on the same patches. The picture is not uniform across families. Llama reads broad moral content; Qwen reads beyond the single harm cue but is unresolved at our sample size; GPT-OSS reads harm, and its refusals can be argued in either direction by its own reasoning trace. Where refusal reads only a low-rank slice and routes around the bulk of what the model knows, a rank-one edit removes it. Whether widening what refusal reads would also deepen the behavior is the open question this raises.

Orion Reblitz-Richardson

Distiller Labs.

Refusals Moral Reasoning Large Language Models Fail Contextual Cues Slices Subspace

Sep 10, 2026eess.SP

Reliable Near-Field Multi-User Positioning Informed by Two-Stage MUSIC

Near-field localization is a promising technique for high-resolution multi-user positioning in future wireless systems, but its performance is often degraded by scattering-induced coherent propagation. Existing near-field localization methods, which require separate parameter estimation and path/source association, suffer from high computation overhead and accumulated errors, and usually do not provide any guarantee on reliability. In this paper, we propose \emph{MUSIC-Net}, an end-to-end near-field positioning deep learning (DL) framework informed by two-stage MUltiple SIgnal Classification (MUSIC) in mixed line-of-sight (LoS) and non-LoS (NLoS) multi-path scenarios, which embeds the two-stage MUSIC objects into training to isolate the LoS-related signal subspace and to identify a surrogate distance. The proposed framework directly recovers multi-user positions without the need for involved NLoS parameter estimation or path/source association. Furthermore, we introduce split conformal prediction (SCP) to move beyond point-estimation-based positioning towards statistically guaranteed (confidence) set estimation for all users. Numerical results show that the proposed MUSIC-Net achieves lower mean positioning error (MPER) than existing benchmarks and yields tighter SCP-calibrated prediction regions, demonstrating both accurate LoS localization and efficient uncertainty quantification (UQ) in coherent multi-path environments.

Jiaying Li, Haifeng Wen, Changsheng You +2

The Hong Kong University of Science and Technology (Guangzhou), Guangzhou, China · Southern University of Science and Technology, Shenzhen, China · The University of Hong Kong, HK SAR, China +1

Indoor Localization Subspace Shortest Paths

Sep 9, 2026cs.CR

From Cycle Space to Cycle Manifold: Limits and Achievability of Blind False Data Injection Attacks

A false data injection attack (FDIA) can change the estimated grid state while evading a residual-based bad data detector (BDD). Existing blind attacks learn a low-rank measurement subspace, but this algebraic view does not state the physical grid constraints that make an attack stealthy or the minimum information needed to recover the complete attack space. Under the connected direct-current (DC) branch-flow model, we show that the residual-sensitive subspace of the noiseless orthogonal test is exactly the weighted cycle space. Its orthogonal complement is therefore the complete stealthy attack space, making weighted cycle-space knowledge both necessary and sufficient for complete blind FDIA. This space identifies the topology only up to 2-isomorphism and the relative cycle-edge parameters only up to one scale per biconnected component; bridge parameters are neither identified nor required. We then formulate a computationally unconstrained benchmark and a tractable measurement-only reconstruction method. Experiments on IEEE systems compare BDD bypass rate at a 95% nominal-acceptance threshold against state impact. As a compact alternating-current (AC) extension, we characterize feasible branch P/Q measurements by a cycle manifold and demonstrate topology-assisted manifold fitting and measurement generation on a graphics processing unit (GPU). In the lossless fixed-voltage small-angle limit, the normal space of the active-power slice reduces to the DC weighted cycle space.

Xin Li, Chenhan Xiao, Jonathan Cohen +3

Faculty of Computer and Information Science, Ben-Gurion-University, Be’er Sheva, Israel · School of Electrical, Computer and Energy Engineering, Arizona State University, AZ, USA · Department of Computer Science, Shamoon College of Engineering, Be’er Sheva, Israel

Real-World Cloud Fault Injection Dataset Subspace Impossibility

Sep 8, 2026cs.CV

A Joint 2D-3D Statistical Shape Model for Orthopedic Reconstruction

Three-dimensional femoral reconstruction from radiographs supports surgical planning, implant sizing, and post-operative follow-up, but remains ill-posed as X-ray projections discard depth information. Existing methods often incorporate a 3D statistical shape model (SSM) as a shape prior to guide reconstructions toward anatomically plausible shapes, relying on iterative 3D-to-2D projection matching. Yet, these approaches are computationally expensive and constrain their SSM to a single dimensionality, leaving the statistical relationship between 2D observations and 3D geometry largely unexploited and unexplored. We instead propose a joint 2D-3D SSM that explicitly captures the co-variation between 2D and 3D segmentations in a shared latent space. During training, 2D and 3D segmentations are registered to a common 3D template and its corresponding 2D projections, and the resulting stationary velocity fields are jointly decomposed using principal component analysis (PCA). This joint modeling allows the 2D-to-3D mapping to be learned directly from data rather than computing correspondences at inference time. For unseen subjects, the 3D shape is recovered directly by lifting the 2D latent coordinates to the 3D PCA subspace, thereby eliminating the need for iterative 3D-to-2D projection. Experiments on NMDID demonstrate that the proposed joint 2D-3D SSM outperforms a widely-used 3D-only SSM baseline while achieving inference approximately 4 times faster, at under 3 seconds per subject. The code is available at: https://github.com/florence-dellaniello-picard/joint2d3d-ssm.

Florence Dell'Aniello Picard, Pranav Poudel, Nairouz Shehata +2

Polytechnique Montréal, Canada · Mila - Quebec AI Institute, Canada · CHUM - University of Montreal Hospital, Canada

3D Reconstruction Statistical Shape Model Feet X-Ray Subspace

Sep 8, 2026cs.LG

SIM: Subspace Interaction-based Method for Token-Level Text Anomaly Detection

Token-level text anomaly detection, as an emerging trend of text anomaly detection, moves beyond coarse-grained document-level detection by localizing anomalous tokens within text. By providing fine-grained abnormality prediction, token-level text anomaly detection plays a critical role in various real-world applications, such as spam filtering and fake news detection. However, existing methods still rely on the global distance calculation for scoring, during which the local anomaly signals are severely diluted by numerous redundant normal feature dimensions. Moreover, pre-trained language models used in these methods inevitably smooth out surface anomalies, further limiting their effectiveness in token-level anomaly detection. To address these limitations, we propose a Subspace Interaction-based Method (SIM for short) for token-level text anomaly detection. To prevent local signal dilution, SIM adopts a subspace interaction-based anomaly detector, which decouples high-dimensional token embeddings into multiple low-dimensional ones, amplifying localized anomaly signals hidden within specific dimensions. To counteract the over-smoothing effect, we design a hard pseudo-anomaly generation module to construct pseudo-anomalous tokens, simulating the subtle anomalies obscured by semantic smoothing. Also, a probabilistic boundary loss is developed to standardize anomaly scores into statistical distances, effectively enforcing anomalous instances to deviate significantly from the normal distribution center. Extensive experiments on multiple benchmark datasets verify the effectiveness of SIM and demonstrate its remarkable efficiency, robustness, and interpretability. The source code is available at: https://github.com/yankehan/SIM-TAD.

Kehan Yan, Yue Tan, Qingfeng Chen +3

Guangxi University, Guangxi, China · Griffith University, Queensland, Australia

Interpretable Anomaly Detection Text Analysis and Detection Subspace Similarity and Metrics

Sep 7, 2026cs.CL

MeRoTune: RoPE-Safe Merging with a Tunable Dial

When you merge two fine-tuned models from the same base checkpoint by simply averaging their weights, you implicitly assume their attention subspaces are still aligned. Recent work attempts to fix misalignments by learning an invertible correction matrix,

M

, for each model's query and key projections. This correction cancels out---using

M

on the query side and

M^{-T}

on the key side---right before the dot product. However, this cancellation is only exact if nothing sits between the projection and the dot product. In reality, almost all modern open-weight language models put a rotary position embedding (RoPE) exactly there. In this paper, we show that this cancellation is exact under RoPE if and only if

M

commutes with RoPE's per-position rotation. We derive the specific class of matrices where this holds: a scaled rotation acting independently within each RoPE frequency pair. This forms a strict, low-dimensional subset of the unconstrained matrices that current methods normally train. Building on this, we turn this constrained matrix class into a new merging method. While keeping the base weights entirely frozen, two fine-tunes each learn their own RoPE-compliant correction matrices. We optimize these corrections against a chosen blend ratio so the final result can be adjusted post-hoc like a dial, rather than locked into a single fixed merge. Our default approach trains at one fixed blend ratio, similar to how LoRA sets its scaling hyperparameter in advance. We also experiment with resampling the blend ratio randomly at every training step, and we report the results of both approaches.

Salman Faroz

Rotary Position Embedding Soft-Token Representations Per-Node Correction Semantic Blending Tuning Dynamic Rope Manipulation+2

Sep 7, 2026cs.IR

EigenLI: Spectral Approximations to Late Interaction

Late-interaction models such as ColBERT achieve strong effectiveness by representing each document with many token-level vectors, but this expressivity leads to large indexing cost, storage footprints and expensive MaxSim scoring. We show that late-interaction representations exhibit an intrinsic low-rank structure: document token embeddings concentrate in a low-dimensional subspace that preserves most of the retrieval signal. Leveraging this observation, we introduce EigenLI, a spectral approximation framework that compresses late-interaction representations via document-specific low-dimensional subspaces. Unlike clustering or pooling methods, EigenLI identifies the dominant eigendirections of each document and uses them to construct reduced interaction representations. Empirically,

k

-EigenLI with

k \le 32

outperforms k-means and Ward clustering based pooling methods on ColBERTv2 and AnswerAI-ColBERT-small; GTE-ModernColBERT exhibits a different tradeoff at

k=32

, where clustering methods perform better. The same spectral construction also yields EigenLI-SV, an ANN-compatible single-vector representation derived from the second-order summary of the reduced structure. Across multiple datasets and all three text models, EigenLI-SV consistently outperforms comparable single-vector surrogates such as MUVERA.

Archish S, Sabyasachi Basu, Ankit Garg +2

Temple · Microsoft Research India

Subspace Spectral Decomposition Spectral Representation Method

Sep 7, 2026cs.LG

Think Wider: Mitigating Latent Rank Collapse in Implicit Chain-of-Thought Reasoning

Chain-of-thought (CoT) reasoning improves the reasoning ability of large language models by introducing intermediate computation, but explicit rationales increase decoding length, latency, and context cost. Implicit CoT offers a more efficient alternative by moving intermediate reasoning into continuous latent states. However, latent reasoning can be unstable: successive latent states may become overly similar and collapse toward a shared dominant direction, reducing the diversity of the reasoning trajectory. In this work, we identify

\textit{latent rank collapse}

and propose

\textbf{WIDER}

, a lightweight spectral regularizer for implicit CoT. During training, WIDER estimates the shared direction of each latent trajectory and penalizes projections onto this direction, encouraging latent states to span a broader representational subspace. The method is plug-and-play and leaves the backbone model, latent schedule, and inference-time decoding procedure unchanged. We further formulate this collapse as a geometric bottleneck in implicit reasoning, casting its mitigation as a training-time regularization problem rather than an inference-time decoding change. Extensive experiments show that WIDER improves matched implicit CoT baselines, while mechanistic analyses reveal higher effective rank, lower dominant-direction energy, and reduced redundancy among latent steps. These results highlight latent subspace utilization as an important factor for efficient continuous reasoning, providing a geometric perspective for analyzing and improving implicit CoT. Code is available at https://github.com/whitesweater/WIDER.

Yuwen Hao, Menglin Yang

The Hong Kong University of Science and Technology (Guangzhou)

Efficient Latent Reasoning Chain-of-Thought Reasoning Intermediate Latent States Collapse Subspace

Sep 7, 2026cs.AI

PLUME: Parameter-Efficient Personalization of Large Language Models via Low-Rank User Modulation in Shared Subspaces

Personalizing large language models (LLMs) is essential for delivering AI assistance that aligns with individual users' styles, intents, and preferences. While per-user fine-tuning can substantially enhance personalization quality, it introduces significant parameter and storage overhead, limiting scalability to large user populations. We propose PLUME (Personalized Low-Rank Adaptation through User Modulation and Shared Subspace), a lightweight framework that achieves efficient and expressive per-user adaptation by leveraging a shared task-specific subspace. Specifically, PLUME first learns a global task subspace from aggregated user data. Personalization is then achieved by training only a lightweight small square matrix within this subspace, enabling each user to obtain a tailored model while keeping shared components fixed. Cross-layer shared parameters and rank-1 residual terms are further introduced to significantly reduce redundancy while maintaining expressiveness. Experiments on multiple personalized text generation benchmarks demonstrate that PLUME achieves comparable or superior performance to strong baselines, while reducing per-user parameters by over 95%. These results establish shared-subspace modulation with minimal residuals as a scalable and semantically grounded approach to LLM personalization.

Xinyu Li, Hao Zhou, Jianfeng Zhu +4

Department of Computer Science Kent State University Kent, OH 44242, USA

Large Language Model Personalization Personalization Subspace

Sep 3, 2026cs.LG

On the Interaction Between Model Compression and Test-Time Adaptation

Deep neural networks deployed in the wild must be both efficient and adaptable, requiring model compression and test-time adaptation (TTA). While both are well studied in isolation, their interaction remains poorly understood. We systematically analyze how structured compression affects a model's ability to adapt under distribution shift. Using ResNet-18 and ViT-Base on CIFAR-10-C and ImageNet-C, we evaluate multiple compression methods combined with standard TTA techniques. We introduce a diagnostic framework that examines representational expressivity and adaptation subspace compatibility. Our results reveal a consistent gap: although compressed models retain high accuracy under supervised adaptation, their TTA performance degrades significantly with increasing compression. We show that this stems from reduced representational diversity and structural constraints that limit recoverability. These effects strongly depend on the compression method, highlighting the need to design compression strategies that preserve adaptability.

Francesco Corti, Dong Wang, Young D. Kwon +2

Graz University of Technology Graz, Austria · Samsung AI Center-Cambridge University of Cambridge Cambridge, United Kingdom · University of Cambridge Cambridge, United Kingdom +1

Stable Test-Time Adaptation Compressed Model Test-Time Adaptation Data Compression Methods Subspace Expressivity

Sep 3, 2026math.AG

Grassmann--Plücker Parametrization of Convolutional Filter Subspaces: Regularity and Closed Embeddings

We propose a geometric parametrization of the filters in a single convolutional layer: the parameter is no longer an ordered family of filter vectors, but a fixed-dimensional subspace of the filter space. For one-dimensional finite-stride convolution, the filter-to-convolution-operator correspondence gives an injective linear map

\mathcal{C}:\mathcal{K}\to H

. This map sends filter subspaces in

\mathrm{Gr}(q,\mathcal{K})

to operator subspaces in

\mathrm{Gr}(q,H)

; composing it with the Plücker embedding yields a projective parametrization

Φ:\mathrm{Gr}(q,\mathcal{K})\to\mathbb{P}(\bigwedge^q H)

. Using

T_U\mathrm{Gr}(q,\mathcal{K})\cong\mathrm{Hom}(U,\mathcal{K}/U)

, we compute the differential of the induced Grassmannian map and show that the differential of

Φ

is injective at every point. We then use the vanishing equations for Plücker coordinates and standard affine coordinates on a Grassmannian to prove that

\mathrm{Gr}(q,\mathcal{C}(\mathcal{K}))\hookrightarrow\mathrm{Gr}(q,H)

is a closed embedding, and hence that

Φ

is a closed embedding. Consequently, the parameter space is isomorphic to its projective image, the parametrization is finite and birational onto its image, every fiber is a singleton, and the resulting projective neural variety is smooth. For

k=4

and

q=2

, we also use Singular to recover the image ideal and check its dimension, degree, chart rank, and smoothness. This computation illustrates, rather than replaces, the general proof. Finally, we discuss possible connections with filter redundancy and low-rank convolution, while distinguishing the proved geometric results from application proposals requiring numerical validation.

Hongyu Yuan, Huaiqing Zuo

Zhili College, Tsinghua University, Beijing, 100084, P. R. China · Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, P. R. China

Subspace Hilbert Series Convolutional Regularity

Sep 2, 2026cs.LG

Routing Is Not Enough: Diagnosing Intra-Adapter Subspace Contention in MoE+LoRA Fine-Tuning

Multi-domain fine-tuning often combines MoE routing with LoRA, assuming that token-level routing separates domain-specific updates. We test this assumption in MoE+LoRA using Python code paired with biomedical text and mathematical reasoning. Although these domains show near-disjoint expert routing, adding biomedical data substantially increases code perplexity, indicating that routing separation alone may not prevent negative transfer. To localize the failure, we introduce Jaccard routing overlap and adapter-gradient cosine similarity, which measure expert sharing and update compatibility, respectively. These diagnostics indicate that interference arises mostly from nearly orthogonal domain gradients competing within the same low-rank adapter subspace. We address this issue with SpawnLoRA, which dynamically adds gated sub-adapters inside MoE experts when adapter-level contention is detected, while keeping the router fixed. We evaluate SpawnLoRA on Phi-tiny-MoE-instruct and OLMoE-1B-7B across multiple mixture settings and find that it effectively reduces negative transfer compared with standard and rank-adaptive LoRA. These results demonstrate that structural separation inside experts provides benefits beyond routing or rank expansion alone.

Mehreen Hossain Chowdhury, Nowshin Mahjabin, Ahmed Shafin Ruhan +3

1Islamic University of Technology · 2York University · 3Dialpad Inc.

Sep 1, 2026cs.LG

Multi-Head Self Attention is a Parameter Identification Mechanism

We prove that a multi-head scaled dot product attention can be viewed as a parameter identification strategy. The ratio of unidentified parameters to the total number of parameters scales like the reciprocal of the number of heads (

1/2 \to 1/(2H)

), meaning models with more heads are structurally more identified. A subtle side effect of the mathematics observation that attention can never be fully identified. Similarly we also show that some bias terms can have no effect on softmax-based attention layers in both the single- and multiple-head settings, though this is mostly a curiosity that should have a marginal effect on model size and model training/prediction efficiency. We also touch on modern improvements to transformers including RoPE and GQA from this perspective, illustrating how those as well can improve the ratio of

meaningful'' parameters to all parameters. Simple numerical examples demonstrate that training can indeed involve updates that overlap model-invariant subspaces that arise from a lack of identification. As part of our experiments we use a

rebalancing'' approach that can ``fix'' updates that overlap unindentified subspaces but do not try to present evidence this should actually be adopted. Instead we simply view our numerical results as exploring and confirming the theoretical results. As a whole we discuss a purely mathematical/statistical explanation, identification, for why specific architectural choices in transformers may have improved performance.

W. Ross Morrow

Multi-Head Attention Linear Attention Subspace

Sep 1, 2026cs.LG

Subspace Levenberg Marquardt Algorithms in Training Neural Networks

The Levenberg-Marquardt (LM) algorithm is a well-known second-order method for rapid convergence and strong robustness when training small- to medium-sized neural networks (NNs). However, its computational and memory costs increase significantly as the number of parameters in an NN grows. To address this limitation, subspace methods have been proposed, such as the Krylov subspace LM (KSLM) and the hybrid subspace LM (HSLM), making second-order algorithms more efficient. In this work, we evaluate the subspace Levenberg-Marquardt algorithms for regression and classification tasks in neural networks. We compare the performance of subspace LM variants with the classical LM method, as well as other popular first-order algorithms, such as stochastic gradient descent (SGD) and Adam.

M. Duc Hoang

Department of Mathematics, University of California, Davis, Davis, CA, USA

Neural Network Training Neural Network Subspace Newton Stochastic Gradient Descent

Sep 1, 2026cs.LG

Frozen Cores Need Task Signal: Fisher-Whitened Cross-Covariance for Low-Resource LLM Adaptation

Parameter-efficient fine-tuning is usually framed as a question of how many parameters to update. Under a severe trainable-state budget, however, where those coefficients act is equally consequential. We study this choice through frozen-core adaptation: a calibration pass fixes left and right bases for each weight matrix, and fine-tuning optimizes only an

r\times r

core. This removes the ability of trainable factors to repair a poor initial span and makes subspace quality directly observable. We introduce FCCA, which estimates the signed input--error cross-covariance, whitens it with diagonal Fisher moments, truncates it in the resulting local metric, maps the selected directions back, and applies thin QR to obtain stable core coordinates. Under a matched

r^2

budget, we compare eight basis constructors on 11 tasks, four model settings, and three seeds. On Qwen2.5-3B, FCCA reaches an 83.0 macro-average, 2.3 points above the next-best matched-budget constructor, and exceeds its unwhitened RawGrad control on all 11 tasks. It ranks first at all three Qwen scales and finishes within 0.13 points of the best method on Llama-3.2-1B. Controlled ablations show gains of 2.7--17.2 points from whitening and identify QR as necessary for stable core optimization in the tested regime. Finally, FCCA comes within 0.32 and 0.23 average points of LoRA and DoRA while optimizing 36.9K rather than roughly 7.4M parameters. These results show that a carefully selected fixed span can recover most of the benefit of movable low-rank factors at a much smaller trainable and optimizer-state cost.

Wentao Ye, Zhanming Shen, Zhiqing Xiao +3

Zhejiang University · Hunyuan, Tencent

Large Language Model Adaptation Frozen Subspace Covariance Fedavg Token Budget Allocation

Sep 1, 2026cs.LG

Breaking the Structural Identity: Personalized Federated LoRA Fine-tuning under Rank Heterogeneity

Large Language Models (LLMs) have achieved remarkable success across diverse domains, but their adaptation to privacy-sensitive, distributed datasets remains a challenge. While Federated Learning (FL) combined with Low-Rank Adaptation (LoRA) provides a resource-efficient paradigm for collaborative fine-tuning, practical deployments are hindered by the dual challenges of resource heterogeneity and data heterogeneity. Existing rank-heterogeneous methods primarily focus on bridging dimension mismatches for aggregation but typically provide a unified global model for all clients sharing the same rank, failing to capture client-specific features in non-IID scenarios. In this paper, we propose FedRoRA (Federated Rank-wise Personalized LoRA), a novel framework that enables fine-grained personalization within rank-heterogeneous federations. FedRoRA decouples adaptation into shared global directions and personalized rank-wise magnitudes governed by learnable diagonal scales. On the server side, it extracts a global subspace via singular value decomposition (SVD) and redistributes client-specific initializations through a personalized projection and top-

k

selection mechanism. Extensive experiments on NLU and NLG benchmarks demonstrate that FedRoRA consistently outperforms state-of-the-art methods.

Lei Wang, Jieming Bian, Letian Zhang +1

University of Florida · Gainesville, FL 32611 · Middle Tennessee State University +1

Federated Learning Model Fine-Tuning Heterogeneity Subspace

Aug 31, 2026cs.LG

PRIME: Mitigating Subgroup Optimization Competition in Shared CTR Top Networks with Plug-in Residual Input-Conditioned Mixture of Expert

Click-through rate (CTR) models vary in feature-interaction design, yet their top networks usually remain a single multilayer perceptron shared by all examples. Heterogeneous user, item, and context subgroups therefore update the same parameters; weakly aligned learning signals make the aggregate gradient a compromise among competing directions. We study the competition on Avazu with 4 models and 4 semantic fields. Across all architectures, semantic subgroups show lower Top-NN gradient cosine similarity than random groups matched by sample size and label ratio, with reductions of 0.23-0.37. This competition motivates input-conditioned experts, but directly replacing an established Dense mapping changes its initial function, sharing pattern, and capacity, obscuring the source of gains. We introduce PRIME (Plug-in Residual Input-conditioned Mixture of Experts), a Dense-anchored mixture of low-rank residual experts. PRIME anchors the original prediction and uses zero-residual initialization to match the Dense baseline exactly at training onset. Input-dependent routing weights low-rank experts for example-specific logit corrections; multi-bag aggregation and EMA load biases stabilize conditional estimation. We evaluate PRIME on held-out Avazu and Criteo test sets across 13 CTR architectures and five paired seeds. Median paired AUC gains are +0.0022 and +0.0066, with LogLoss reductions of 0.0011 and 0.0081, respectively. On FiBiNET and DCNv2, PRIME outperforms APG in all ten seed-level AUC comparisons while using fewer parameters and lower inference latency on both backbones. These results show that function-preserving conditional residuals add input-dependent capacity while preserving the Dense path and its optimization stability. Code is available at https://github.com/YH-learning/PRIME.

Heng Yao, Siyun Hou, Tianying Liu +8

Ant Group · Henan Polytechnic University · Independent Researcher +1

Click-Through Rate Prediction Subspace Dense Baseline Experts Receiver Operating Characteristic Area Under The Curve Inference Latency

Aug 31, 2026math.ST

Compact and Infinite-Order Error Analysis for Null-Space SVD Estimation

We study null-space estimation from a noisy matrix. For a simple left null space, we first derive an exact compact expression for the error of the smallest left singular vector. We then give an all-order series for the SVD vector and projector, followed by compact and consistently truncated series forms for the fixed-realization empirical risk and conditional population generalization risk. The recursion extends to a multiple-dimensional null space by following the complete invariant subspace. The convergence radius is not inferred from an error plot: it is computed independently from the nearest complex exceptional point that joins a retained eigenvalue branch to its complement. A reduced-nullity experiment shows that moving this spectral boundary can increase the radius, although the improvement is not monotone in the retained nullity. For individually ordered null directions under Gaussian training with

τ\geq m

, we prove that the Wishart splitting matrix

W

gives a strict second-order empirical ranking. Gaussian averaging equalizes the leading generalization risks at both small and very large noise, while a column-swap theorem proves strict expected generalization ranking for an isotropic signal subspace. For unequal spikes, an exact population-overlap criterion and a simultaneous

99\%

Monte Carlo confidence certificate explain the observed intermediate ranking. A sixth-order risk correction improves the lower-crossover estimate in the reported experiment. This equal--ranked--equal phenomenon is a finite-sample diagnostic related to spectral mixing, but its tolerance crossings, the exceptional-point radius, and the asymptotic BBP threshold are three distinct quantities.

Xin Li, Jonathan Cohen, Rami Puzis

Singular Value Decomposition Subspace Singularity

Aug 28, 2026cs.LG

Intrinsic Interaction Geometry Controls the Low-Rank Complexity of Softmax Attention

How much matrix rank is required to preserve every bounded value output of normalized softmax attention? We study the unrestricted maximum-row-

\ell_1

approximation rank

r_\varepsilon(A)

, exactly the least rank achieving uniform error over all bounded vector-valued values. Row softmax exposes the intrinsic interaction

C=P_m(\log A)P_N

, whereas invertible

Q/K

gauges leave

A

fixed while changing the Euclidean geometry of a chosen query/key factorization. We replace that coordinate-dependent description by a projective residual

q(C-T)

and an attained factor-radius size

κ(T)

. For every rank-

r

retained interaction with

τ(T)<\varepsilon

, we prove

r_\varepsilon(A)\le \min\left\{ N,\; C_r\left( 1+\frac{κ(T)} {(\varepsilon-τ(T))^2} \right)^{r/2} \right\},

with the same unknown dimension constant as the underlying weighted Gibbs-row cover. The profile is gauge invariant, termwise no worse than native retained-subspace bounds at the same declared dimension, and has a worst-case sharp

r/2

size exponent at fixed

r

and

\varepsilon

. We then measure

r_\varepsilon(A)

directly on learned attention using 9,978 certified brackets across BERT, GPT-2, Qwen2.5, and two ViT checkpoints; where certificates do not close, the optimum remains interval-valued. A pre-specified 2,302-cell held-out study further shows that the historical native-coordinate geometry block contains coarse, mostly head-level information but no detectable incremental information beyond a strong calibrated baseline. The new intrinsic descriptor is not evaluated in that study. Together, the theory and measurements distinguish an operator-intrinsic complexity control from a stronger empirical explanation that the learned-head evidence does not support.

Yuhe Sui, Jianing Zhang, Yingzhi Tang

Nanyang Technological University, Singapore · Carnegie Mellon University

Low-Rank Structure Spectral Norm Gumbel-Softmax Relaxation Subspace

Aug 16, 2026cs.LG

SubZero+: Memory-Efficient Adaptive Zeroth-Order LLM Fine-Tuning in Random Subspaces

Zeroth-order (ZO) optimization with SGD in random subspaces enables memory-efficient fine-tuning of large language models without backpropagation. However, high gradient estimation noise fundamentally undermines adaptive optimizers like Adam. We propose SubZero+, which achieves practical adaptive ZO optimization through a carefully designed dual low-dimensionality strategy: (i) multi-query forward-difference gradient estimation in periodically refreshed random subspaces to mitigate noise amplification in moment buffers, and (ii) Adam updates with periodic restarts performed directly in low-dimensional space rather than full-parameter space. In experiments, this dual design retains memory overhead comparable to momentum-free ZO methods while achieving stronger optimization performance than the evaluated ZO baselines. Theoretically, in the exact-directional limit,

K

-query averaging preserves conditional unbiasedness, while the coefficient estimator's covariance and mean-squared error, as well as query-induced second-moment inflation, scale exactly as

1/K

. Extensive experiments across SuperGLUE with models from 1.3B to 32B parameters under both full fine-tuning and LoRA schemes demonstrate consistent improvements over competing ZO methods. SubZero+ significantly narrows the performance gap with first-order optimization while preserving ZO's inference-time memory efficiency.

Ziming Yu, Shuyao Xiao, Xingyu Zhao +6

Beijing Normal University · Singapore Management University · Xiaomi Inc. +2

Zeroth-Order Large Language Model Fine-Tuning Subspace Bounded-Memory

Aug 13, 2026cs.CV

SCOPE: Subspace Clustering with Online Per-Head Top-K Estimation for Sparse Video Attention

Diffusion Transformers (DiTs) incur quadratic self-attention cost over spatiotemporal tokens. Existing training-free sparse attention methods often construct sparse masks from block-level or cluster-level proxy scores, which can obscure fine-grained differences among keys and miss high contribution keys under aggressive sparsity. Moreover, such proxy scores may yield overly concentrated softmax distributions, causing Top-

p

to retain too few keys for some query clusters. Although a fixed Top-

k

minimum alleviates this failure mode, a shared value cannot adapt to variations across heads and inputs. To address both limitations, we propose SCOPE, a training-free sparse attention framework that combines 3D-RoPE-aligned key subspace clustering with online per-head Top-

k

estimation for efficient video-DiT inference. SCOPE partitions post-RoPE keys into temporal, height, and width subspaces, clusters them independently, and aggregates the corresponding centroid scores through lookup tables to obtain per key proxy scores for each query cluster. Building on existing hybrid Top-

p

/fixed Top-

k

selection, SCOPE derives a head-specific Top-

k

value online by averaging the initial retained key counts within each head, weighted by query cluster size, and selects additional keys only for query clusters whose initial retained key counts fall below this value. Sparse attention is then computed over the selected original keys and values. Across six model--task configurations, SCOPE consistently outperforms existing training-free baselines in both fidelity and latency, achieving up to a

1.99\times

end-to-end speedup on 720p HunyuanVideo with

28.46

dB PSNR relative to dense attention.

Qi Zhao, Qirui Li, Hanlin Tang +10

Zhejiang University · Alibaba Group

Dynamic Sparse Attention Spatio-Temporal Attention Subspace Top-K

Aug 12, 2026cs.LG

High-dimensional Multi-objective Bayesian Optimization with Learned Variable Interactions

Multi-objective Bayesian optimization (MOBO) is effective in identifying the Pareto fronts for expensive black-box problems. However, most current MOBO approaches are limited to low-dimensional decision space due to its exponential sampling complexity. This paper presents decision variable interaction analysis-based MOBO, ViaMOBO, a generic framework for expensive multi-objective problems with high-dimensional decision space. The key idea of ViaMOBO is that it utilizes a variable interaction analysis model to determine whether the decision space can be completely or partially divided, and then performs local Bayesian optimization in the divided decision subspaces. Through the variable analysis model, it can be derived whether the objectives in black-box problems are separable, partially separable, or non-separable based on the potential independent or interdependent relationships among decision variables without any strong assumptions. We compare ViaMOBO with the state-of-the-art MOBO methods on both synthetic and real-world benchmarks. The experimental results demonstrate that ViaMOBO outperforms other related MOBO baselines in approximating the Pareto front of high-dimensional expensive multi-objective problems.

Hongyan Wang, Jiayu Huang, Haotian Zheng +6

School of Vehicle and Mobility Tsinghua University Beijing, China · Faculty of Engineering The Hong Kong Polytechnic University Hong Kong, China

Multi-Objective Bayesian Optimization Algorithms Multi-Objective Optimization High-Dimensional Subspace