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12 papers in the last four weeks, up 50% on the four weeks before. 0.1% of all new papers.

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Latest papers 71

Oct 1, 2026cs.SI

Degree-Corrected Joint Matrix Factorization for Multilayer Community Detection

Multilayer networks allow the modeling of interactions between the same entities across different contexts, such as temporal observations, varying settings, or interactions of different types. The goal of community detection in multilayer networks is to identify groups of nodes exhibiting similar connectivity patterns, which may vary across layers. We propose a method based on a joint nonnegative symmetric matrix trifactorization for community detection in multilayer networks, where each graph is approximated by a nonnegative symmetric matrix trifactorization. Our approach enforces constraints on the factor matrices so that communities are disjoint and shared across layers, while allowing each layer to have its own connectivity patterns and node degrees. This flexibility enables the model to capture both local and global structural variations across layers. We also develop an algorithm to efficiently solve this problem. We evaluate multilayer community detection methods using the multilayer degree-corrected stochastic block model (MDCBM), a flexible framework for generating realistic multilayer graphs with heterogeneous degrees and varying connectivity patterns. Experiments show that our method reliably detects communities across diverse regimes, whereas existing state-of-the-art approaches are often limited by restrictive structural assumptions.

Alexandra Dache, Manon Rustin, Arnaud Vandaele +1

Department of Mathematics and Operational Research, University of Mons, Mons, Belgium

Community Detection Network Science Restricted Bernoulli Matrix Factorization Matrix

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

Beyond Accuracy: Prefix-Invariant Realizations of Low-Precision Fast Matrix Multiplication

Fast matrix multiplication saves multiplications through exact cancellation, but rounding sums that mix token rows can leave contributions from later tokens in earlier language model outputs. This threatens prefix invariance, which multiple-choice likelihood scoring relies on: a scored likelihood must depend only on its allowed prefix. On Qwen2.5-14B-Instruct, two fast FP8 realizations repaired to ordinary-looking accuracy still change the answers chosen by likelihood on 5.83% and 10.00% of 240 OpenBookQA items when only the text after the allowed prefix is replaced with the bf16 model's own greedy continuation. Both row-local controls, the bf16 model and a deployed FP8 matrix multiplication kernel, change none. Accuracy thus does not certify prefix invariance, and the stability criteria we analyze cannot tell realizations apart: across all 512 sign variants of two-level Strassen they stay constant while teacher-forced perplexities span a 772.4

\times

range on the same model. We therefore construct certified realizations of two-level Strassen on bounded integer codes that quantize token rows independently, then mix and cancel exactly before rescaling, using 49 block multiplications instead of 64. Our certificate guarantees bitwise equality to a prescribed row-local classical int8 operator at the same quantization specification, so every certified realization inherits its prefix invariance. Certification thus turns realization choice into a pure cost decision: which certified realization runs can no longer change a single scored likelihood.

Shuxiao Xie, Shuyang Xie, Yuan Cao +3

Matrix Multiplication Quantized Matrix Uncertainty Score

Sep 30, 2026cs.LG

MatrixReward: Reward from Rubric Matrix for Open-Ended Generation

Open-ended query generation lacks standard answers, thus necessitating an effective reward mechanism. Pointwise scoring rubrics provide limited information about the relative quality of sample answers under the same prompt; merging multiple rubric judgments into a single score may also mask the differences between these answers. We propose MatrixReward, which constructs rewards from a rollout-by-rubric win-rate matrix obtained by comparing every pair of sampled responses under each rubric. The spread of each matrix column captures how strongly that rubric distinguishes the current rollouts, while correlations between columns reveal rubric repetition; together, these statistics yield data-dependent rubric weights. We combine these weights with the prior weights of rubrics. After column normalization and weighting, the observed per-rubric maxima and minima define positive and negative ideal profiles. Each rollout's distances to these two ideals determine its relative-closeness quality reward. Evaluated using Qwen3-8B on four open-ended query-answering benchmarks, MatrixReward achieves an average score of 63.02, outperforming the strongest baseline by approximately 2.0%. These results support the idea that matrices derived from relative comparisons can be used to construct rewards more reasonably for open-ended generative reinforcement learning.

Zihan Shen, Qi Liu, Zixuan Yang +4

Zhejiang University · Qwen Business Unit of Alibaba · Renmin University of China

Reward Signal Reasoning Benchmark Offline Reinforcement Learning Matrix

Sep 30, 2026cs.IR

A Shared Taste for Model-Written Text: The Generator-by-Selector Matrices of "AI-AI Bias" Show No Detectable Own-Model Premium

Laurito et al. (PNAS 2025) showed that large language models choosing between two descriptions of the same product, paper or film prefer the description written by a language model over the one written by a person, by a wide margin over what human judges do. Their design crosses five generators with the same five models as selectors, which permits a second question the paper does not headline: does a selector prefer text from its own model beyond what the generator and selector main effects predict? We rebuild the three 5x5 matrices from the per-item counts in the authors' public repository (21,828 valid trials; every cell matches the published value) and fit a two-way fixed-effects model with an own-model term gamma, tested by the exact permutation test over the 120 relabellings of the selectors. The premium is +0.013 on products (exact one-sided p = 0.24), -0.010 on paper abstracts (p = 0.74), +0.054 on films (p = 0.07) and +0.019 pooled (p = 0.14; 95% interval -0.008 to 0.046). The same-vendor term for the GPT-3.5 and GPT-4 pair is negative in all three datasets. Position bias moves single cells by up to 0.42 share points in either direction, and the own-model contrast is unchanged once order-driven items are removed. The design would have detected a premium of 0.05 with 82% (products), 88% (papers), 42% (films) and 97% (pooled) power; the minimum detectable effect at 80% power is 0.034 pooled. The absence is informative down to about 0.04 share points and silent below that. The 4x4 matrix of Tan et al. (ACL 2024) gives gamma = +0.148 at the smallest p its 24 relabellings allow, with a same-family term of the same size. The main result of Laurito et al. stands: models share a taste for model-written text, with GPT-4's descriptions chosen 77% to 95% of the time by every selector on products. What these data do not show is a model recognising and favouring its own prose.

Dmitrij Żatuchin

Department of Information Technologies, Estonian Entrepreneurship University of Applied Sciences (EUAS), Tallinn, Estonia. · Rankfor.AI, Wroc law, Poland; Tallinn, Estonia.

Large Language Model Bias Human-Written Text Position Bias Contrastive Learning Matrix

Sep 28, 2026cs.LG

Price Stability in the European Union: A Systemic Approach Using Random Matrix Theory

Price stability remains a pillar in monetary policy practices and carries a special importance within monetary unions. Mainstream economics tried to leverage price stability using price indices and several metrics to shed light on specific dynamics and optimal macroeconomic levels. The wide availability of data led researchers to consider the study of systems using Random Matrix Theory, based on inner correlation patterns. This aims to enhance the multivariate analysis by removing noisy patterns from the signal and improve data quality for further inferences. This work considers the collection of monthly inflation indices in the Eurozone as a \textit{system} of prices to analyze its eigenvalues' statistical and asymptotic properties and uncover inner country-level insights. Results confirm the system cannot assumed to be randomly generated, and the data exhibit noise-dominated patterns, due to small and persistent variations at the country-level. The latter make the inter-country correlations more dynamic and the separation of the signal from the noise quiet difficult. Findings identified two countries as distorting inflation dynamics besides three other distinct, regional-based groups of countries. Variability sources might stem from economic episodes fueling inflation spikes in some countries, as well as methodological aspects used to ensure data quality and representativeness in the European Union. Despite being complex, the system demonstrates a certain stability, in terms of self-organization; while large monthly fluctuations cannot be considered as rare events, but part of the data-generating process.

Sami Diaf

Department of Socioeconomics Universität Hamburg

Sep 24, 2026stat.ML

Nuclear Norm-Regularized Bayesian Matrix Completion

Matrix completion, the problem of estimating missing entries in a matrix from noisily observed ones, underlies a diverse array of problems such as recommender systems and counterfactual outcome estimation in panel data. Many algorithms address the problem using regularized least squares, often with the nuclear norm as a regularizer, but this method yields a point estimate with no built-in uncertainty quantification. A Bayesian formulation is a natural alternative, and if the noise variance is known, the nuclear norm-based prior yields a log-concave posterior. Unfortunately, in practice, the noise variance will not be known a priori, so for a fully Bayesian approach, a prior must be imposed on it. We give the first sampler for this model with an explicit non-asymptotic guarantee: polynomial in the matrix dimensions and in the reciprocal of the target accuracy. Our technique is to discretize the distribution of the noise precision onto a grid and build a categorical posterior via thermodynamic integration. This extension is not specific to matrix completion and may be useful in other non-log-concave sampling problems where the non-log-concavity is restricted to a single variable and the joint distribution of the remaining variables is nonsmooth. Our contribution is a feasibility result: we show that a polynomial-time Bayesian sampler for this model exists at all, and the resulting complexity, while polynomial, is not intended as a deployable algorithm at current problem scales.

Calvin Tolbert

School of Operations Research and Information Engineering Cornell University

Tensor Completion Matrix Log-Concave Distributions Spectral Norm Completion Nuclear Fusion

Sep 24, 2026cs.LG

Growth-Inspired Graph Generation and Inverse Design of Mechanical Lattices via Dot Matrices Database Augmentation and GCNN

Natural load-bearing and transport networks are not assembled in a single step; they emerge through a temporally ordered process of growth, branching, reinforcement, and loop formation. Inspired by this developmental logic, this work introduces a morphogenetic graph-generation framework for mechanical lattices in which a discrete dot matrix provides potential nodes and the final architecture is created by sequential cross-layer and intra-layer growth. The same rule is visualized in two dimensions as a leaf-vein-like developmental sequence and implemented in three dimensions on a 3x3x3 nodal matrix containing 27 candidate nodes. A dataset of distinct three-dimensional lattices was evaluated by beam-based finite element analysis and represented directly as graphs. A graph convolutional neural network (GCNN) with three graph-convolution layers and dual global pooling learns the topology-property mapping and predicts effective compressive stiffness. Coupling the GCNN surrogate with rapid structural sampling enables inverse design: for a target stiffness of 1000 MPa, the selected design was predicted at 1042.43 MPa and validated by finite element analysis at 1027.49 MPa. Beyond straight members, the framework has also been extended to parameterized horseshoe-shaped curved beams made of nonlinear materials, enabling topology-geometry design toward prescribed deformation shapes. Our work provides a paradigm for augmenting the database of mechanical metamaterials, and the resulting perspective links biological morphogenesis, graph learning, and nonlinear shape programming in a unified generative design framework for architected materials.

Weiyun Xu, Jiamu Liu

Department of Civil and Environmental Engineering, University of Illinois Urbana-Champaign, Urbana, Illinois 61801, USA · State Key Laboratory of Clean and Efficient Turbomachinery Power Equipment, Department of Mechanical Engineering, Tsinghua University, Beijing, 100084, China

Topology Optimization Finite Element Method Lattice Matrix Data Generation Inverse Design+1

Sep 23, 2026cs.LG

Riemannian Structure and Optimization for a Class of Low-Parametric Orthogonal Matrices

In this paper, we are concerned with matrices formed by block-diagonal factors interleaved with fixed permutations -- a flexible family of structured matrices. This class has recently drawn interest in deep learning architectures for its balanced expressivity-efficiency trade-off, yet efficient computational strategies for working with it remain to be found. We approach this problem through Riemannian geometry and examine under what conditions this class admits a smooth manifold structure. For the practically important case of orthogonal two-factor matrices, we derive the essential Riemannian tools and propose efficient algorithms for their implementation. The algorithms leverage automatic differentiation, support parameter sharing within each factor, and avoid explicit dense matrix construction. We test them within the Riemannian optimization framework on the best matrix approximation problem and for parameter-efficient fine-tuning of large language models. Beyond the two-factor setting, we study the geometric and matrix-theoretic properties of factorizations with a larger number of block-diagonal factors.

Ali Aliev, Maxim Rakhuba

HSE University

Riemannian Optimization Orthogonality Riemannian Manifolds Matrix Manifolds Deep Learning

Sep 23, 2026stat.ML

Matrix Aggregation Operators

Aggregation theory has traditionally focused on operators defined over vectors. However, many applications-including Multi-Criteria Decision Making, Group Decision Making, Fuzzy Rule-Based Classification Systems, and overlap/grouping indices-require aggregating information naturally structured as a matrix of membership degrees (e.g., where a set of objects interacts with a family of fuzzy sets). Despite this, no formal framework has been proposed for this class of operators, partly due to the common practice of flattening matrices into vectors (which discards structural information) and partly due to a reliance on decomposable operators that aggregate rows and columns sequentially. This paper addresses this gap by formalizing the notion of a matrix aggregation operator (MAO). We analyze the decomposability and symmetry properties of MAOs, showing that certain operators cannot be expressed in decomposable form and examining several notions of symmetry. Finally, we introduce a family of MAOs termed maximum entropy global coverage indices (MEGCIs), provide a construction method for them based on combining grouping functions and MEOWA operators, and illustrate their usefulness in cluster quality assessment through an extensive computational study.

Inmaculada Gutiérrez, Asier Urio-Larrea, J. Tinguaro Rodríguez +3

Faculty of Statistical Studies, Complutense University of Madrid · Instituto Universitario de Estadística y Ciencia de Datos, Complutense University of Madrid · Department of Statistics, Computer Science and Mathematics, Universidad Pública de Navarra +4

Matrix Secure Aggregation Clustering Fuzzy

Sep 16, 2026cs.LG

Randomized SVD Approximations for Spectral Co-Clustering of Word-Document Matrices

Spectral co-clustering is a useful tool for discovering latent structure in word-document matrices, but its reliance on singular value decomposition (SVD) can make standard formulations expensive on high-dimensional data. This paper presents two randomized approximations for normalized spectral co-clustering of bipartite text data when the numbers of document and word clusters may differ. The first method uses randomized SVD through random projection, while the second combines partial SVD with element-wise random sampling. Across real-world and synthetic datasets, both methods reduce runtime relative to the full-SVD baseline, but their behavior depends on matrix sparsity. The random projection method is the more reliable approximation across the tested settings, whereas the sampling-based method is most useful on denser matrices and provides limited benefit on already sparse text data. These results show that randomized approximations for spectral co-clustering should be selected according to the underlying structure of the data.

Fateme Mazdarani, Carlos Toxtli

School of Computing Clemson University Clemson, SC, USA

Singular Value Decomposition Clustering Matrix Approximation Latent Variable

Sep 14, 2026stat.ML

Data Attribution at Scale via Influence Matrix Estimation

Data attribution seeks to quantify how individual training examples shape a model's predictions and underpins problems including data valuation, machine unlearning, and model interpretability. Despite having a long line of work, computationally scalable methods often struggle to predict the effect of removing training data in neural networks due to their non-convex nature. To overcome this challenge, metagradient-based methods such as MAGIC (Ilyas and Engstrom, 2025) differentiate each prediction through the entire training run and compute its exact influence with respect to the training data, but require a separate run for every prediction. To reduce this cost, we cast budgeted attribution as estimating a large influence matrix from a small number of measurements. We show that the measurements most appropriate for recovering this matrix differ from those best suited for attribution itself. We then present two algorithms, MAGE and SPELL, suited for reconstruction and attribution respectively, that run on existing metagradient machinery at no extra cost. Empirical studies demonstrate strong performance over existing baselines across training scales and measurement budgets.

Yuxi Chen, Hamza Golubovic, Han Tong +2

Carnegie Mellon University 1Department of Statistics & Data Science · Columbia University 2Department of Statistics · Columbia University 3Electrical Engineering Department +2

Influence Function Influence Interpretability Matrix

Sep 9, 2026quant-ph

A Quantum-Inspired Dequantization Method for Diagonally Weighted Matrix Functions: Application to Learning with Optimized Random Features

Quantum-inspired classical algorithms have dequantized several quantum machine learning routines by replacing quantum linear-algebra subroutines with classical counterparts. However, the sampler based on quantum singular value transformation (QSVT) for learning with optimized random features is not covered by existing dequantization frameworks, because the matrix to be inverted is not itself available through sampling access. In this work, we develop a classical algorithm to address this type of quantum-advantage candidate. Our method samples heavy indices, reduces the transformation to a small principal block, and outputs a sparse classical representation with operator-norm guarantees. Applying this method dequantizes the sampler for optimized random features, giving a classical sampler with prescribed accuracy and polynomially related runtime. These results show that the factorization underlying a quantum block encoding can itself provide sufficient classical structure even when sampling-and-query access to the composite matrix is unavailable.

Natsuto Isogai, Mio Murao, Hayata Yamasaki

Department of Physics, Graduate School of Science, The University of Tokyo, Tokyo, Japan · Department of Computer Science, Graduate School of Information Science and Technology, The University of Tokyo, Tokyo, Japan

Quantum Computational Advantage Atom-Averaged Features Singular Value Decomposition Quantum Error Correction Matrix Markovian Sampling

Sep 3, 2026eess.SY

Formation Matrix and Energy-based Control of Multi-Agent Systems

This paper presents an energy-based controller for a multiagent robotic system designed to achieve and maintain a specific formation while moving on a plane and avoiding collisions between agents. The controller emulates a network of elementary spring-damper modules connecting pairs of agents. This network, with its de-energized states representing the desired formation, determines the system's dynamics, which is fully encapsulated by a bond graph model. The modeling is further enhanced through the introduction of a formation matrix, using a graph-theoretic approach, that describes both the distances and relative velocities among the agents of the arrangement. This matrix mathematically represents the interconnection and energy-exchange structure of the bond graph, allowing us to put it in correspondence with the control-by-interconnection CbI-scheme of the IDA-PBC theory, facilitating the solution of the formation control problem within the port-Hamiltonian system framework. Furthermore, the paper presents leader-following and position-based formation control systems based on the CbI scheme, including a stability analysis of the corresponding closed-loop systems. The theoretical findings are validated through numerical simulations across various scenarios.

Martín Crespo, Sergio Junco, Matías Nacusse

Automation and Control Systems Laboratory - FCEIA, National University of Rosario, Argentine

Multi-Robot Systems Optimistic Follow-The-Regularized-Leader Hamiltonian Matrix

Sep 2, 2026cs.LG

The Gradient Does Not See Rank: Rank-Indifference in Matrix-CODI on ProsQA

Continuous chain-of-thought models compress reasoning into latent tokens. Matrix-valued variants, which route each latent token through a d x d matrix bottleneck, introduce rank as a single-sample structural observable on the latent matrix Z. If matrix latents carry parallel reasoning paths via superposition, rank should track them, and truncating Z to low rank should hurt accuracy on tasks whose solutions plausibly require multiple components. Across four training regimes of a matrix-CODI model (three on ProsQA, one on GSM8K-Aug below the learning threshold), the rank-k projection ablation curve is flat to within 0.6 percentage points. A three-seed replication yields 81.0 +/- 2.0 percentage points accuracy while the final effective rank of Z spans {4, 12, 13}; the loss does not reward any particular rank. To test whether rank-blindness arises from the flatten-then-project readout alone, we trained four readouts: a bilinear reparametrization, a bilinear-plus-GELU readout nonlinear in Z, an SVD-augmented readout feeding singular values through an MLP, and a quadratic readout in Z Z^T. All four rank-k curves remain flat (Spearman p-values 0.63, 0.14, 0.82, 0.46). The flat curves persist for readouts nonlinear in Z. A linear probe on Z underperforms a raw pretrained hidden state at target prediction (AUC 0.673 vs. 0.846). A negative control on vanilla GPT-2 SFT (no matrix bottleneck, no Z, three seeds, n=500) reproduces a flat rank-k curve under the same intervention paradigm with pooled-mean range 0.20pp, and a random-h sensitivity floor lands at the same accuracy: the rank-k ablation alone conflates rank-blindness with position-irrelevance.

Samuel Larson

Pebble ML

Low-Rank Structure Matrix Ranking Ablation Readout Gradient+1

Sep 1, 2026cond-mat.mtrl-sci

Text-guided flow matching enables sample-efficient crystal structure generation

Crystal generators can now propose periodic structures, but their control interfaces remain poorly matched to the mixed descriptors used in materials design. Text provides a compact way to combine composition, symmetry, prototype and property cues, yet it has not been clear whether such information can steer flow-based crystal generation. Here we introduce TFMat, a text-conditioned flow-matching framework that uses structured materials language as a semantic prior for a CrystalFlow generator. Across Perov-5, Carbon-24 and MP-20 crystal structure prediction benchmarks, TFMat improves one-candidate match rates over CrystalFlow and reaches a 92.04% MP-20 match rate with 20 candidates; in de novo generation, it improves element-count and density distribution alignment while retaining coarse property consistency in composition-selected outputs. These results position structured text as an inspectable control layer for translating human-readable materials intent into candidate crystals for downstream simulation and validation.

Wentao Li

1Tsinghua University

Crystal Structure Crystal Structure Prediction Molecular Generation High-Fidelity Conditional Generation Conditional Flow Matching Matrix

Aug 31, 2026cs.LG

Learning Materials Properties from Scarce Labels and Unlabeled Crystals

Learning materials properties from scarce labels and unlabeled crystals is a central challenge for data-driven materials discovery. We present SemiMat, a controlled benchmark for semi-supervised materials property regression, and MatRank, a reliability-weighted objective for continuous pseudo-label uncertainty. SemiMat fixes labeled and unlabeled crystal inputs, graph-backbone interfaces, validation-only checkpoint selection, held-out test reporting, normalized MAE (NMAE), and method-rank summaries across six scarce-label tasks, four graph backbones, and five predefined split runs. MatRank builds pseudo-targets from labeled anchors, weights them by local reliability and weak-prediction agreement, trains weak and strong graph views consistently, and adds ranking signals so that unlabeled crystals shape both values and candidate order. Across the retained 24 backbone-task blocks, one fixed MatRank objective gives the lowest aggregate held-out test NMAE (0.896) and best average method rank (2.208). The component, OOD, and generated-pool diagnostics identify where the gain is reliable and where further screening evaluation remains necessary. Code is available at https://github.com/littlepeachs/SemiMat.

Wentao Li, Yizhe Chen, Jiangjie Qiu +3

1Beijing Key Laboratory of Artificial Intelligence for Advanced Chemical Engineering Materials, State Key Laboratory of Chemical Engineering and Low-Carbon Technology, Department of Chemical Engineering, Tsinghua University

Data-Driven Materials Discovery Materials Crystal Matrix

Aug 31, 2026eess.SP

Benchmarking External Generalization of SPD Matrix Learning for Resting-State fMRI Connectome Prediction

Resting-state functional magnetic resonance imaging (rs-fMRI) functional connectivity (FC) matrices are widely used for individual-level prediction, but strong performance within one cohort may not generalize to a new cohort. We ask whether within-dataset performance remains when the test data come from an entirely held-out rs-fMRI dataset. Each scan is represented as a regularized symmetric positive definite (SPD) correlation connectome, which allows methods to use the geometry of the SPD manifold. We introduce a reproducible age-prediction benchmark across six rs-fMRI datasets: COBRE, ADNIDOD, Cam-CAN, ABIDE, OASIS-3, and ADNI. The benchmark compares a vectorized correlation baseline, Tangent-Space Ridge, SPDNet, and split-wise Riemannian harmonization under within-dataset GroupKFold, pooled GroupKFold, and leave-one-dataset-out (LODO) evaluation. Within-dataset and pooled GroupKFold results are substantially more favorable than LODO results. When an entire dataset is held out, prediction error increases, differences among methods narrow, and performance is strongly affected by age-range mismatch and cohort heterogeneity. The benchmark provides common inputs, model settings, data splits, and analysis scripts so that future SPD matrix learning methods can be evaluated under the same external-validation protocol.

Ce Ju, Antoine Collas, Florent Bouchard +1

Inria, CEA, Universit´e Paris-Saclay, Palaiseau, France · Universit´e Paris-Saclay, CNRS, CentraleSup´elec, L2S, France

Human Connectome Project Full-Rank Correlation Matrices Vascular Age Kernel Ridge Regression Matrix

Aug 12, 2026cs.LG

Exploring Oversmoothing with Householder Matrices

Deep graph neural networks(GNNs) suffer from oversmoothing- a progressive collapse of node representation towards a low information subspace as network depth increases because the normalized graph propagation operator is repeatedly applied directly to the hidden representations. In this work we study Householder Graph Neural Network (HouseGNN). Rather than updating the hidden state like standard GCN, HouseGNN uses the aggregated neighbourhood message solely to estimate a reflection direction; the node embedding is then updated by a Householder reflector followed by GroupSort, yielding a piecewise orthogonal layer that preserves Euclidean norm at every node and at every depth. We prove three core properties: (i) every internal layer preserves the node-wise Euclidean norm; (ii) the Householder reflector is scale scale and sign-invariant in the message; and (iii) pairwise distance between nodes can change through mismatch between node-wise orthogonal operators.

Bhaskar Karol

Graph Neural Networks Spectral Norm Matrix

Aug 12, 2026cs.DS

Fast Length-Squared Sampling for Positive-Semidefinite Matrices

We describe a simple rejection-sampling-based algorithm to perform length-squared sampling on an

n \times n

positive-semidefinite (psd) matrix: that is, to sample a column with probability proportional to its squared

\ell_2

-norm. The algorithm runs in just

O(n)

expected time, which is significantly sublinear in the input matrix size. The runtime is optimal, even when the input is assumed to be diagonal. Our result has several applications. Length-squared sampling is used by a number of sublinear time algorithms for matrix problems, like low-rank approximation and eigenvalue approximation. Often, it is assumed that the algorithm is given access to the matrix column norms, and thus can perform length-squared sampling efficiently. Our result shows that, at least for psd matrices, we can remove this assumption. We also discuss an application to an asymptotically optimal algorithm for estimating the Frobenius norm of a psd matrix to relative error. Finally, we show that our sampling algorithm yields a very simple sublinear time algorithm for the robust psd low-rank approximation problem introduced by Bakshi et al. (FOCS, 2020), which nearly matches the more complex method developed there.

Rajarshi Bhattacharjee, Ethan N. Epperly, Cameron Musco +1

University of Massachusetts Amherst · University of California Berkeley

Matrix Low-Rank Structure Optimal Sample Complexity Convex Relaxation Rejection Sampling

Aug 12, 2026cs.LG

MOON: Multi-Objective OrthoNormalized Updates for Multitask Learning

Multi-objective optimization (MOO) has demonstrated significant success in multi-task learning by mitigating task conflicts through gradient manipulation. However, most existing methods flatten model parameters into vectors and perform gradient manipulation under Euclidean geometry, thereby overlooking the matrix structure prevalent in modern architectures such as Transformers. In this paper, we show that gradient manipulation in Euclidean space does not generally yield the steepest descent direction under matrix geometry, potentially limiting optimization efficiency. Drawing from the theory of steepest descent for matrix-valued parameters, we propose MOON (Multi-Objective OrthoNormalized Updates), which performs gradient manipulation under spectral--nuclear norm geometry and uses the orthonormalized manipulated gradient for parameter updates. Theoretically, for smooth non-convex objectives, we establish convergence of the averaged Pareto-stationarity measure at rates of

\mathcal{O}(T^{-1/2})

in the deterministic setting and

\mathcal{O}(T^{-1/4})

under stochastic gradients. Empirical results across various benchmarks show that MOON consistently improves both optimization efficiency and final multi-task performance. Our code is available at https://github.com/KunlinLyu/MOON.

Shiji Zhou, Kunlin Lyu, Lei Zhang +2

Institute of Artificial Intelligence, Beihang University · Beijing Advanced Innovation Center for Future Blockchain and Privacy Computing, Beihang University · Beijing Academy of Artificial Intelligence (BAAI) +1

Multi--Task Learning Multi-Objective Optimization Matrix

Aug 12, 2026cs.CV

New Orthogonal Multiwavelet Filters Derived by Matrix Spectral Factorization

The paper considers the construction of two new orthogonal multiwavelets with supercompact support by using the Fast Bauer's method for matrix spectral factorization on the matrix product filter of the orthogonal CL multiwavelet filter. The new multiwavelets possess orthogonality, symmetry/antisymmetry, and one of them provides better coding and smoothness than other supercompact multiwavelets. The performance of the new multiwavelet filters in subband-based edge detection, grayscale and color image compression and 1D and 2D signal denoising is compared with the GHM, SA4, CL, Integer Haar and Alpert multifilters. The comparative analysis shows that new multiwavelets can provides better human visual measures, SSIM and MS-SSIM in image compression and denoising applications.

Vasil Kolev, Todor Cooklev, Fritz Keinert

Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Bl. 2, Acad. G. Bonchev St., 1113 Sofia, Bulgaria · Wireless Technology Center, Purdue University, Fort Wayne, IN 46805, USA · Department of Mathematics, Iowa State University, Ames, IA 50011, USA

Wavelets Spectral Filtering Matrix

Aug 11, 2026cs.CL

Seeds Before Objectives: Rethinking Evaluation for Low-Resource Garhwali ASR

At corpus sizes typical of low-resource dialects, single-run comparisons can yield gains that do not replicate. We show this for Garhwali, an under-resourced Indo-Aryan language of the central Himalaya, building the first reproducible multi-seed ASR benchmark on the official VAANI splits, with per-seed outputs and significance testing. Re-examining plausible gains, we find them fragile: neither Focal CTC nor a matra-weighted objective beats standard CTC under seed-level testing, the matra objective fails to cut even its targeted errors, and Hindi-to-Garhwali transfer gives no gain over direct fine-tuning. What holds up is mundane: w2v-BERT 2.0 with standard CTC reaches 47.0% WER over five seeds, beating the larger MMS-1B and comparable models; pretraining design, not parameter count, drives performance, and speed augmentation gives a small, largely consistent gain. Multi-seed evaluation on official splits separates real gains from seed noise.

Karamvir Singh Batra, Prathamjyot Singh, Ashima Sood +2

Thapar Institute of Engineering and Technology, Patiala, Punjab, India · Ulster University, Londonderry, United Kingdom · Ulster University, Belfast, United Kingdom

Automatic Speech Recognition Evaluation Low-Resource Languages Automatic Speech Recognition Text Corpora Objectives Matrix

Aug 11, 2026physics.optics

Causality Sum Rules in Conventional Scattering Matrices

Scattering matrices are the standard experimental and computational description of photonic and electromagnetic devices. Passivity is explicit in the conventional incoming-outgoing matrix, whereas causality sum rules are usually formulated only after transforming the response into auxiliary variables. Here we show that these rules can be written directly in the conventional scattering matrix by removing the time advance introduced by the reference domain. Using the earliest-arrival delay of each channel, we define a domain-delayed matrix that preserves real-frequency passivity while restoring the causal time origin. Under explicit analyticity, transparency, and regularity assumptions, this matrix becomes a Schur function, enabling a Cayley-Herglotz construction. The resulting projected and determinant bounds constrain coherent channel superpositions and aggregate multichannel loss. The framework recovers Rozanov's absorber limit and spherical-multipole sum rules, while extending causality bounds to measurable quantities including insertion loss, suppressed singular-value channels, and conditional lossless delay-bandwidth trade-offs. Our work directly connects fundamental causality theory with experimentally accessible scattering data. The initial theoretical route is autonomously explored by Qiushi Engine, an AI research system for open-ended scientific discovery, and subsequently verified, refined, and developed by the authors, demonstrating a hybrid AI-human discovery workflow.

Ning Han, Rui Zhao, Shuxing Yang +3

Interdisciplinary Center for Quantum Information, State Key Laboratory of Extreme Photonics and Instrumentation, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Zhejiang University, Hangzhou 310027, China · International Joint Innovation Center, The Electromagnetics Academy at Zhejiang University, Zhejiang University, Haining 314400, China · College of Optical and Electronic Technology, China Jiliang University, Hangzhou 310018, China +2

Matrix Active Flow Control

Aug 5, 2026cs.LG

Matrix Zonotopic Attention: A Context-Adaptive Value Projection for Set Transformers

Multi-head attention combines an input-dependent softmax routing with an input-independent linear value projection, so the per-sample operator mapping aggregated values to outputs is the same for every input set. We study the consequences of this asymmetry for permutation-invariant set targets. We introduce the Transformation Degrees of Freedom (TDOF) of a target operator, a complexity measure counting the input-dependent directions an exact representation requires, and present a depth-separation analysis showing that context-rigid attention needs depth proportional to the target's TDOF, whereas a single layer with a context-adaptive value family can represent the same target. Building on this analysis, we propose Matrix Zonotopic Attention (MZAttn), which replaces the fixed value projection with a context-adaptive matrix-zonotope family: a centre matrix plus a sum of generator matrices weighted by input-dependent gates. The construction reduces to standard multi-head attention at initialisation, preserves permutation equivariance, and admits a data-driven reachability interpretation. Experiments on a range of set-prediction tasks are consistent with the TDOF prediction that the architectural advantage is selective: it appears on targets that depend on the input set in a high-rank, sparsely combinatorial way, and is small on aggregate-statistic targets where parameter-matched standard attention is already competitive.

Zhen Zhang, Amr Alanwar

School of Computation, Information and Technology Technical University of Munich, Germany

Multi-Head Attention Transformer Architectures Gumbel-Softmax Relaxation Matrix Polytopes

Aug 4, 2026cs.AI

MatrAIx: Simulating the World with 8.3 Billion Persona Agents

Human evaluation of AI systems and digital products is costly, slow, and difficult to scale. Offline evaluations are more scalable but often abstract away human diversity and interactive behavior. We therefore introduce MatrAIx, a population-scale simulated-user evaluation infrastructure for testing AI systems and digital products with heterogeneous users. MatrAIx has three core components: First, Persona 8B contains 8.3 billion persona records represented by 1,290 categorical dimensions. Records are either sampled from a dependency graph that preserves correlated attributes or derived from human-authored profiles. We release a quality-filtered coreset of approximately 1 million personas, comprising 599,847 human-grounded and 400,000 synthetic records. Second, the MatrAIx Playground provides four environments in which diverse users evaluate and interact with digital products: Survey, AI Chatbot, Web, and App. Third, MatrAIx provides 1,010 application tasks spanning more than 25 domains, including Commerce, Software, Finance, and Healthcare. We conducted 18,189 evaluation trials across eight representative tasks. Persona agents were powered by three LLMs: Claude Opus 4.8, GPT 5.5, and Claude Haiku 4.5. The resulting feedback captures how decisions and preferences vary across persona backgrounds, including hesitation after a price increase, willingness to continue after an AI assistant fails, and latency tolerance. We conducted two main validation studies: First, a 400-trial controlled study evaluated persona adherence across ten behavioral attributes and all four environments. The declared behavior was expressed or correctly suppressed in 366 trials (91.5%). Second, human and LLM judges evaluated the extraction quality of human-grounded personas. Overall, MatrAIx provides an end-to-end infrastructure for evaluating AI systems and digital products with diverse simulated human users.

Xiaomin Li, Yuexing Hao, Jianheng Hou +90

User Simulation Persona Consistency Matrix World

Jul 30, 2026physics.data-an

Rethinking Total Absorption Gamma Spectroscopy Deconvolution: Supervised Machine Learning vs Response-Matrix Methods

The extraction of

β

-feeding distributions in Total Absorption

γ

-ray Spectroscopy constitutes a challenging inverse problem, particularly in nuclei with complex decay schemes involving a large number of excited states. In such cases, the measured spectrum arises from the superposition of many detector response functions, making the determination of the individual feedings intrinsically ill-posed and highly sensitive to the methodology employed. In this work, we present a systematic comparison between supervised Machine-Learning techniques and Response-Matrix methods using realistic Monte Carlo simulations of an experimental Total Absorption Spectrometer. Supervised Machine-Learning approaches construct a non-parametric estimator that infers level feedings from the measured spectrum after a training stage, whereas Response-Matrix methods determine the feeding distribution by directly minimizing the difference between measured and reconstructed spectra. Our results show that supervised Machine-Learning techniques achieve superior accuracy in the reconstruction of individual feeding intensities, whereas Response-Matrix methods provide robust and physically consistent initial solutions. These findings support a hybrid strategy in which a Response-Matrix method is first used to obtain an initial feeding estimate, which is then refined using a supervised Machine-Learning approach to achieve improved overall accuracy.

J. Balibrea-Correa, E. N{á}cher, C. Fonseca-Vargas +1

Instituto de Física Corpuscular, CSIC - Universidad de Valencia, Spain, 46980, Valencia, Spain

Cosmic-Ray Syntactic Pool Laser-Induced Breakdown Spectroscopy Semi-Supervised Learning Matrix Distributions

Jul 29, 2026cond-mat.mtrl-sci

MatCreatioNN: Machine learning-guided computational discovery of photocatalysts for environmental applications

The rational design of photocatalysts for environmental remediation and CO2 conversion remains limited by the high computational cost and sparse experimental data describing multi-parameter photocatalytic behavior. This work presents an integrated machine-learning framework that couples reinforcement learning-based metal-organic framework (MOF) generation with a multi-stage Crystal Graph Convolutional Neural Network (CGCNN) prediction funnel to identify photocatalysts optimized across multiple electronic and structural features. 120,000 MOF candidates were generated and screened using 13 key descriptors, including band-gap suitability, CO2/H2O selectivity, adsorption energy, and structural stability. The funnel approach reduced computational cost by 4.13-fold while maintaining predictive robustness. Two top candidates, a Cr-based and a Zn-based MOF, exhibited predicted photocatalytic fitness values of 1.70 +/- 0.25 and 1.20 +/- 0.05 fold higher respectively than benchmark materials such as PCN-224(Zr), demonstrating simultaneous improvements in light absorption, redox energetics, and framework durability. Simulated X-ray diffraction patterns confirmed strong structural agreement with experimentally synthesized MOFs, indicating high synthesizability. Post-hoc analysis revealed recurring structural motifs, such as the N262 metal cluster, that correlated strongly with high predicted photocatalytic activity. These results highlight the potential of data-driven methods to accelerate discovery of efficient and durable photocatalysts for environmental and energy-related transformations, providing a foundation for experimental realization and large-scale implementation of computationally designed MOFs.

Satya Kokonda

Department of Chemistry, School of Engineering, Charter School of Wilmington, 100 N Dupont Rd, Wilmington, DE 19807, United States of America

Heterogeneous Catalysis Matrix Environment

Jul 27, 2026cs.LG

Low-Rank Dependence Decomposition via Accelerated Symmetric Non-negative Matrix Factorization

Symmetric non-negative matrix factorization (SymNMF) recovers latent group structure from a dependence matrix, but its dense, quadratic-memory objective has confined prior work to moderate sizes. We present a large-scale GPU study of seven algorithm families (over 30 configurations) on absolute Pearson correlation and tail pairwise dependence matrices from Extreme Value Theory, two proxies for empirical risk-factor estimation on large portfolios. A trace-identity reformulation eliminates all

n \times n

intermediates, so a single GPU reaches

n \approx 10^5

and multi-node distribution scales to

n = 10^6

and beyond. Under a two-phase protocol, eleven methods converge at moderate scale; six remain efficient enough at

n = 10^5

(five AdaGrad-family plus ADMM), and five AdaGrad-family methods still converge at

n = 10^6

: AdaGrad, RMSprop, and three we introduce (Piecewise AdaGrad, Row-Stochastic SVRG, Block-SVRG AdaptGrow). At

n = 10^6

the fastest solver tracks the matrix spectrum: Block-SVRG AdaptGrow wins on the flat, ill-conditioned tail-dependence spectrum, where its lower per-iteration cost decides a long factorization, and full-batch AdaGrad wins on the dominant-low-rank correlation spectrum, where the run is short. We also benchmark spherical K-means as a hard-label baseline: cheaper when angular cluster structure is present, yet provably degenerate once the matrix collapses toward a single common factor, where the soft factorization remains necessary.

Lavinia Ghita, Dhruv Desai, Jake Goldberg +1

NVIDIA

Nonnegative Matrix Factorization Matrix Basis Decomposition Acceleration

Jul 27, 2026cs.CV

MATS: A novel multi-modality multi-task learning framework for 3D perception in autonomous driving

Multi-modality data from different sensors provides rich complementary information for 3D perception, becoming an essential component in reliable autonomous driving systems. Current research typically designs intricate and complex fusion strategies to integrate information from multimodal data on a unified bird's-eye-view (BEV) feature map for the joint learning of multiple perception tasks. However, such a single feature map hardly carries sufficient information to simultaneously meet the requirements of various perception tasks, leading to a very limited perception performance. To mitigate this limitation, this paper proposes MATS, a novel multi-modality multi-task learning approach with modality-adaptive BEV fusion and task-specific Mixture-of-Experts (MoE) for 3D perception. Specifically, a simple modality-adaptive BEV fusion module is designed to adaptively recalibrate the BEV features by modeling the global cross-modality dependencies, generating diverse BEV feature maps for various perception tasks. For joint multi-task learning, this paper proposes a task-specific MoE module to decouple the tasks and enable the network to automatically choose the appropriate BEV feature candidates for each specific task. To validate the effectiveness of the proposed approach, we conduct extensive experiments on the large-scale benchmark nuScenes. With the camera- and LiDAR-modality input data, the proposed approach outperforms the state-of-the-art (SOTA) by a significant margin. Furthermore, the experimental results on the single tasks show that the proposed approach significantly outperforms the baselines. The code and trained models will be available upon publication.

Junchen Huo, Wanming Hao, Song Wang +3

School of Electrical and Information Engineering, Zhengzhou University, Zhengzhou 450001, China · Tianping College of Suzhou University of Science and Technology, Suzhou 215009, China · department of computer science, Toronto Metropolitan University, Toronto M5B2K3, Canada

Autonomous Driving Perception Multi-Sensor Fusion Bird'S-Eye View Autonomous Driving Multimodal Fusion 3D Perception+2

Jul 27, 2026cs.CV

Structural Loss Metrics for Tensor Approximation via Matrix Low-Rank Approximation

Matricized low-rank approximation via SVD is a standard surrogate for tensor decompositions, but entry-wise reconstruction error fails to capture multiway geometric degradation. Under an orthogonal Tucker model, we characterize this degradation using two metrics: cross-mode Direction Loss, measuring geometric subspace deviation from rank truncation and noise rotation, and Interaction Loss, quantifying multilinear interaction distortion in the core tensor. We prove that squared relative reconstruction error orthogonally decomposes into interaction loss and out-of-subspace energy, and derive a Wedin-type bound establishing the stability of a plug-in Direction Loss estimator. Experiments on synthetic and hyperspectral datasets demonstrate that nearly identical reconstruction errors can yield markedly different structural-loss profiles; hyperspectral patches with comparable reconstruction errors exhibit up to a 4.6-fold difference in Direction Loss, correlating with severe visual blurring.

Hiroki Hasegawa

Graduate School of Science and Technology, University of Tsukuba, Tsukuba, 305-8573, Japan

Low-Rank Structure Reconstruction Error Matrix Approximation High-Order Correlations

Jul 24, 2026stat.ML

Graph-Based Correlation Matrix Generation: A Convex Optimization Approach

This work addresses the generation of theoretical correlation matrices with prescribed sparsity patterns associated to graph structures. We propose a novel convex optimization framework in which an initial matrix is projected onto an elliptope under a positive semidefiniteness constraint. Several numerical schemes are implemented and compared. The problem falls within the broader class of matrix completion, where off-diagonal entries corresponding to absent edges are fixed to zero and diagonal entries are fixed to one. Beyond this structural constraint, the approach offers greater flexibility than existing methods by allowing control over the mean of the off-diagonal entry distribution, enabling the generation of correlation matrices that better reflect realistic data. This procedure is not designed to yield a uniform distribution over the feasible set; rather, it provides a principled and tunable way to construct correlation matrices suitable for benchmarking statistical methods for graphical model inference. Theoretical guarantees on the existence of solutions are established, both in the general setting and under the additional mean constraint. Simulation studies illustrate the properties of the generated matrices with respect to graph structure. The methodology is applied to two real-world datasets from neuroscience and finance, and a comparison with GAN-based correlation matrix generation is provided.

Ali Fakhar, K{é}vin Polisano, Ir{è}ne Gannaz +1

UGA · Univ. Grenoble Alpes, CNRS, Grenoble INP, Inria, LJK, F-38000 Grenoble, France · G-SCOP\_GROG, G-SCOP, Grenoble INP, UGA +2

Full-Rank Correlation Matrices Matrix Probabilistic Graphical Models Convex Optimization Spurious Correlations Sparsity

Jul 20, 2026q-bio.GN

Making Single-Cell Data Distillation Auditable: Traceable Real-Cell Coresets via Discrete Min--Max Selection

Large single-cell datasets are expensive to store, curate, and repeatedly reuse for model training. Data distillation can reduce this burden by building smaller training sets. However, many existing methods rely on synthetic cells. These synthetic cells do not retain direct correspondence with assayed cells and genes. This limits source-level inspection and biological traceability. Moreover, real-cell expression matrices are often sparse and noisy. In light of these challenges, we propose Minmax-CF, a label-aware characteristic-function selector for traceable single-cell data distillation. Minmax-CF formulates compression as a discrete min--max selection problem over characteristic-function directions. It uses entropy-regularized maximization to emphasize the least preserved directions. Greedy minimization ranks cells and genes by how much they reduce the resulting weighted error. The method alternates cell and gene selection under explicit axis-specific budgets. Across five coarse-lineage benchmarks and five compression budgets, Minmax-CF retains 95.3% of the Full-reference macro-F1 on average, with gaps that exceed one per-seed standard deviation. It also retains exact source-cell indices and original gene symbols. Compared with size-matched synthetic PCA-Centroid and Distribution Matching (DM) baselines, Minmax-CF achieves higher coarse-lineage macro-F1 in 24 of 25 comparisons against each baseline. It exceeds their average performance by 10.4% and 17.4%, respectively. Retained cells can also be projected onto independently computed embeddings for direct biological interpretation.

Yaodi Luo, Peize He, Lingbei Meng +4

The Chinese University of Hong Kong, Shenzhen · Shenzhen Loop Area Institute, Shenzhen · Shenzhen Research Institute of Big Data, Shenzhen, China

Diffusion-Based Dataset Distillation Geometry-Aware Uncertainty Coresets Probe-Logit Distillation Matrix

Jul 20, 2026cs.CV

Direct Clinical Joint Angle Extraction from Parametric Body Model Rotation Matrices

Quantitative joint angles are rarely available in routine care because the tools are slow, costly, or confined to a laboratory. We show that clinical joint angles can be read directly from the per-segment rotation matrices a parametric body model already produces, with no inverse-kinematics or musculoskeletal-model fitting step. On the OpenCap LabValidation cohort, using the GEM-X body-model estimator on single-smartphone video, our pooled mean absolute error is 4.50 degrees over the fifteen joint angles that match the OpenCap Monocular reference set, the same accuracy range as OpenCap Monocular's 4.8 degrees on the same cohort and reference standard, from a much simpler pipeline. The step that connects a body model to clinical angles is a small calibration table rather than an optimisation, so the same procedure transfers unchanged to other body models: repeating it on SAM 3D Body, changing only the table, gives 4.66 degrees, statistically indistinguishable from GEM-X, and runs in real time from a live single-camera stream. The method needs no per-recording inputs beyond the video itself: no participant height, no camera-intrinsics database, no per-subject model scaling. This broadens where movement analysis is practical, from in-clinic and at-home recording to telerehabilitation and large-scale decentralised studies.

Joey Páolo Kardolus, Daan Hendriks, Jaap Jansen

Babon Innovations B.V., Utrecht, The Netherlands · Research Group Lifestyle and Health, HU University of Applied Sciences Utrecht, The Netherlands

Uncertain Joint Stiffness Body Motion Capture Matrix

Jul 18, 2026cs.LG

TVGL-CFM:Generating and Forecasting Time-Varying Trajectories of Dynamic Networks with Conditional Flow Matching

Many complex systems such as brain networks, financial markets, and gene-regulatory circuits are described not by a fixed graph but by one that changes over time. A standard way to summarise such structure at each instant is the sparse precision (inverse-covariance) matrix, and the time-varying graphical lasso (TVGL) turns a multivariate signal into a smooth chain of these matrices. We introduce TVGL-CFM, a single model that learns the distribution of such chains and can both generate new, realistic time-varying network trajectories for a given class and forecast how an observed trajectory will continue. Each precision matrix lives on a curved space of positive-definite matrices, but a log-Euclidean chart flattens an entire trajectory into an ordinary vector space, so a simple conditional flow-matching model can be trained and sampled there while every decoded matrix is guaranteed to be a valid precision matrix. For forecasting we start the flow not from noise but from a rough extrapolation of the recent history, so the model only has to learn a small correction. Across EEG motor-imagery, chaotic systems, and gene-expression data, TVGL-CFM generates trajectories that keep the class-discriminative structure of real data, and it forecasts future connectivity more accurately than raw-signal baselines. Generating the structured precision trajectory directly is therefore more faithful than generating raw signals and estimating connectivity afterwards.

Om Roy, Yashar Moshfeghi, Keith Malcolm Smith

University of Strathclyde

Conditional Flow Matching Matrix

Jul 15, 2026cs.LG

Local Additive Feature Attribution: A Mathematical Taxonomy and Reporting Checklist

Feature-attribution methods are central to explainable artificial intelligence. Their assumptions are expressed in several mathematical languages: cooperative-game values, path integrals, gradient operators, perturbation distributions, and backpropagation rules. This survey proposes a common framework for local additive feature attribution. It organizes Shapley, path-based, gradient/backpropagation, perturbation, and CAM-style methods around five specification choices: value function, reference, path, perturbation distribution, and conservation rule. It then compares these methods through an axiom-by-method matrix and links common failure modes, including baseline sensitivity, off-manifold perturbations, sanity-check failures, adversarial manipulation, and method disagreement, to the assumptions that produce them. Finally, the survey proposes a ten-item reporting checklist for studies that use local additive attributions. The central message is that attribution results are meaningful only relative to the mathematical assumptions under which they are defined, and that those assumptions should be reported.

Rebecca Afriyie Sarpong, Daniel Commey

Shapley Additive Explanations Explainable Artificial Intelligence Contributions Report Taxonomy Backpropagation+1

Jul 13, 2026cs.CV

A Nearable Soft Mat Based on Distributed Optical Fiber Sensing for Physiological Monitoring

Distributed optical fiber sensing (DOFS) combines the advantages of fiber optic sensors, including flexibility, small size, immunity to electromagnetic interference, and high metrological performance, with the capability to transform a single optical fiber into a continuous sensing element for spatially resolved mechanical measurements. Optical frequency domain reflectometry (OFDR), based on Rayleigh backscattering, enables high spatial resolution DOFS measurements, broadening the range of potential sensing applications. However, OFDR based DOFS remains largely unexplored for biomedical applications, despite the need for sensitive, spatially resolved, and conformable sensing interfaces. This study presents a soft DOFS based mat as a large-area interface for physiological monitoring. A single-mode optical fiber was embedded in a flexible silicone matrix and arranged in a serpentine layout to distribute sensing over the mat surface. With a gage pitch of 2.6 mm, the system provided 2250 sensing sites across the active area at a sampling frequency of 50 Hz. The mat was assessed on six healthy volunteers in a seated nearable configuration on the backrest of a standard office chair. The distributed output enabled two dimensional mapping of the mat response, reflecting back mat mechanical coupling and cardiorespiratory induced perturbations. Respiratory rate and heart rate were therefore estimated and compared with a reference wearable system. The maps revealed physiologically coherent spatial and temporal patterns, while the estimated rates showed good agreement with the reference measurements. These results demonstrate the feasibility of combining large area distributed sensing, spatial mapping, and quantitative cardiorespiratory monitoring within a DOFS based soft nearable interface.

Vincenzo Lavorgna, Martina Pulcinelli, Andrea Polimadei +5

Unit of Measurements and Biomedical Instrumentation, Department of Engineering, Università Campus Bio-Medico di Roma, 00128 Rome, Italy · Photonics Micro- and Nanostructures Laboratory, ENEA Research Center of Frascati, 00044 Frascati, Italy · Fondazione Policlinico Universitario Campus Bio-Medico, 00128 Rome, Italy

Jul 10, 2026cs.LG

Graph-Regularized Low-Rank Matrix Completion by Variable Projection

We address the low-rank matrix completion problem by incorporating graph regularization into the existing Riemannian Trust-Region Matrix Completion (RTRMC) framework. The latter uses the geometry of the low-rank constraint to remodel the problem as an unconstrained optimization problem on a single Grassmann manifold. Our approach, named Graph-Regularized RTRMC (GR-RTRMC), exploits the inherent relationships between rows and columns of the matrix. By using these relationships, we aim to improve the accuracy and robustness of matrix completion, particularly in scenarios where the underlying data exhibits strong correlations between rows or columns.

Benoît Loucheur, P. -A. Absil, Michel Journée

ICTEAM Institute, UCLouvain, 1348 Louvain-la-Neuve, Belgium. · Department of Climatology, Royal Meteorological Institute of Belgium, 1180 Uccle, Belgium.

Tensor Completion Low-Rank Structure Matrix Regularization Projection Completion

Jul 9, 2026cs.LG

MatBind: A Shared Embedding Space for Multimodal Materials Characterization

Fully characterizing a crystalline material requires integrating heterogeneous data sources -- atomic structures, diffraction patterns, electronic density of states, and natural language -- each of which captures a different facet of the same physical object. In practice, however, these modalities are stored and analyzed in isolation, making it difficult to relate or query materials across representational boundaries. We present MatBind, a contrastive learning framework that aligns four materials modalities -- crystal structure, powder X-ray diffraction (pXRD) simulated from structures, density of states (DOS), and text -- into a unified embedding space using crystal structure as the central physical anchor. The framework induces alignment between modalities never explicitly paired during training, enabling emergent zero-shot cross-modal retrieval as a direct consequence of the shared representation. The learned embedding space organizes materials according to physically meaningful properties without explicit supervision, and retrieval performance improves systematically when modalities are combined at query time. These results demonstrate that treating heterogeneous materials data as complementary projections of a single physical reality, rather than as isolated data sources, is not a practical choice but is consistent with the underlying physics.

Le Yang, Anoop K. Chandran, Jona Östreicher +8

Institute for Advanced Simulations · Institute for Advanced Simulations (IAS-9), Forschungszentrum Jülich GmbH, 52425 Jülich, Germany · Jülich Supercomputing Centre, Forschungszentrum Jülich +15

Materials Science Powder X-Ray Diffraction Materials Crystal Structure Contrastive Learning Matrix

Jul 8, 2026cs.CV

A Generalized Deep Non-negative Matrix Factorization Approach for SAR Automatic Target Recognition

The deep nonnegative matrix factorization (DNMF) technique is proposed to address the low interpretability of deep learning-based methods in extracting multilayer features from synthetic aperture radar (SAR) target samples. However, existing DNMF methods employ a layer-by-layer decomposition strategy, which is prone to causing error accumulation and local optimum, thereby hindering a consistent improvement in recognition accuracy as the number of layer increases. In this paper, a robust multilayer feature extraction method, termed generalized deep non-negative matrix factorization (G-DNMF), is proposed to address the above challenges in SAR automatic target recognition (ATR). The G-DNMF aims global optimality and derives the update rules for each parameter using lagrangian multiplier method. The new update formula indicates that both the DNMF method based on the encoding matrix and the mixing matrix are special cases of the proposed method, theoretically demonstrating the universality of proposed method. In general, the proposed method discards the layer-by-layer decomposition strategy, thereby effectively mitigating the risk of local optima and eliminating error accumulation, leading to a significant improvement in DNMF's multi-layer feature extraction capability. The experimental results, by presenting the feature images extracted from each layer by G-DNMF and the reconstructed original images, verified the proposed method's pure additive understanding of multi-layer features and demonstrated its interpretability. The experimental results based on MSTAR and OpenSARship datasets show that G-DNMF outperforms existing DNMF algorithms and their derivatives in terms of stability and recognition performance.

Yunhong Zhang, Changjie Cao, Zhongli Zhou +4

Synthetic Aperture Radar Nonnegative Matrix Factorization Feature Extraction Matrix

Jul 7, 2026stat.ML

The Regularization Parameter: Sparse Precision Matrix Estimation

Sparse precision matrix estimation provides an interpretable and computationally efficient framework for modeling conditional dependencies in high-dimensional, low-sample-size data. A recurring challenge is appropriately selecting the regularization parameter that controls estimator sparsity and strikes a balance between underfitting and overfitting. We propose a closed-form, matrix-valued regularization parameter derived from the sampling distribution of the first-order optimality conditions of the

\ell_1

-regularized Gaussian maximum-likelihood estimator. By prescribing the probability that each nonzero entry of the estimator satisfies its optimality condition under resampling, we eliminate the need for cross-validation. The resulting regularization parameter is shown to attain asymptotic scaling properties that, under standard conditions, provide consistency and sparsistency of the estimator. On synthetic Gaussian and non-Gaussian datasets, as well as real-world gene microarray and neuroimaging applications, the proposed approach achieves estimation accuracy comparable to cross-validation, delivers superior support recovery, and reduces runtime by several orders of magnitude.

Aryan Eftekhari, Daniel Sergio Vega, Ernst-Jan Camiel Wit +1

Institute of Computing, Faculty of Informatics, Universit`a della Svizzera italiana (USI), Lugano, Switzerland

Maximum Likelihood Regularization Matrix Cross Validation Genome-Wide Association Study

Jul 6, 2026cs.LG

CollabEval: Statistically Efficient Collaborative Model Evaluation via Matrix Completion

Evaluating generative AI models is a routine, but resource-intensive, process that is conducted over and over again during the course of model development. In this work, we propose Collaborative Evaluation (CollabEval), a simple, effective, and principled method for exploiting dependencies between historical runs of different models on the same tasks to improve statistical efficiency. Specifically, our approach treats model evaluation as a matrix completion problem over an

M \times N

matrix of evaluation scores, where

M

is the total number of models and

N

is the total number of evaluation prompts. We assume that a subset of these

M

models are targeted for evaluation. For these target models only a small fraction,

p

, of prompts has been annotated with evaluation scores. Leveraging recent results in prediction-powered inference, we build a low-rank approximation of the score matrix, and use the reconstructed values as control variates in a manner that guarantees unbiased estimates of the true evaluation metric mean, in addition to statistically valid confidence intervals. Empirically, across a wide range of datasets, models, and sparsity levels

p

, we find that CollabEval substantially reduces the mean confidence interval size, and the mean squared error of the point estimate, compared to baseline methods at the same annotation budget.

Adam Fisch, Daniel Deutsch, Joshua Maynez +5

Google DeepMind · Google Research

Model Evaluation Multi-Dimensional Evaluation Tensor Completion Matrix

Jul 3, 2026math.PR

An AI-Assisted Solution to the Signed BAR Conjecture: Uniqueness in the Harrison--Reiman Class and a Completely- $\mathcal{S}$ Class Obstruction

For a multidimensional reflected diffusion, determining whether the associated basic adjoint relationship (BAR) uniquely characterizes the stationary distribution is a basic uniqueness problem in the BAR approach. The problem has remained unresolved for more than 35 years since the introduction of the BAR approach. In this paper, we resolve the finite-signed uniqueness problem for stable Harrison--Reiman data with a nonsingular

M

-matrix reflection matrix. The proof uses pathwise differentiability of the reflected diffusion implies feasible directional differentiability of the probabilistic resolvent to show that, at boundary points, its one-sided initial-state derivative factors through the tangent projection and vanishes along active reflection directions. An interior one-sided convolution then yields smooth test functions whose oblique derivatives are uniformly bounded and converge pointwise to zero on each closed face. The interior signed measure is consequently invariant for the reflected semigroup. The proof was discovered with the assistance of ChatGPT 5.5 Pro and subsequently verified by the authors. We also show that the nonsingular

M

-matrix assumption is structural. In the larger completely-

\mathcal{S}

class, a nonsingular reflection matrix with a singular proper principal block admits boundary gauges supported on lower-dimensional strata. Under standard exponential ergodicity and a mild one-step regulator bound, these gauges produce nonzero zero-mass signed BAR tuples; indeed the zero-mass interior BAR coordinates contain an infinite-dimensional subspace. A four-parameter three-dimensional family, including an explicit rational example, verifies the obstruction. Thus the finite signed version of the Dai--Dieker question has a positive answer in the Harrison--Reiman

M

-matrix class and a negative answer in a natural completely-

\mathcal{S}

extension.

Yiping Lu, Youheng Zhu

Department of Industrial Engineering and Management Sciences, McCormick School of Engineering, Northwestern University.

Matrix Adjoint Matching Implicit Differentiation

Jul 1, 2026cs.LG

Token Geometry

Language models learn continuous programs over discrete symbols, with the embedding table and LM-head acting as the read/write interface between them. We show that this interface has gradient geometry distinct from dense hidden weights which can be exploited to improve the Pareto frontier across supervised finetuning, RL, and pretraining, while only utilizing kilobytes of optimizer state. We introduce Ember, a lightweight optimizer for embedding and LM-head matrices that utilizes O(V + D) VRAM, instead of Adam's O(2VD), and forgoes the need to shard both token table optimizer states. We provide empirical evidence that Ember scales effectively across batch size and parameter count. We show that the optimization trajectory of tokens can be well described by a simple 1D ray, counter to the popular belief that neural net parameters navigate a heavily nonconvex landscape. We provide a principled view on the surprisingly narrow space of optimizers that suffice for Transformer training. Finally, we open-source our distributed Ember implementation that merges cleanly with existing ZeRO/FSDP setups to support further research at https://github.com/katop1234/ember

Kathan Shah

Preprint.

Parameter-Efficient Fine-Tuning Methods Matrix

Jun 29, 2026cs.LG

Muon learns balanced solutions in matrix factorization without slow saddle-to-saddle dynamics

Matrix factorization (i.e., problems of the form

\min_{\mathbf{P},\mathbf{Q}} \|\mathbf{M}^\star - \mathbf{P}^\top\mathbf{Q}\|_\mathrm{F}^2

) is a minimal learning problem that exhibits both nonlinear parameter dynamics and representation learning. In this setting, we study how parameter trajectories under the Muon optimizer differ from those of gradient descent. We identify three main dynamical differences: 1) Muon avoids the slow saddle-to-saddle dynamics from small initialization. Muon instead learns all the top modes of

\mathbf{M}^\star

at the same rate, with the smaller modes converging first. 2) Muon remains stable even when the learning rate exceeds the critical threshold set by the local loss sharpness. This frees the learning rate from the condition number of the problem, enabling rapid convergence via exponential learning rate annealing. 3) Once the weights are aligned with each other and the target, Muon flow conserves the matrix quantity

\sqrt{\mathbf{P}^\top \mathbf{P}}-\sqrt{\mathbf{Q}^\top \mathbf{Q}}

, while gradient flow is known to conserve the matrix

\mathbf{P}^\top\mathbf{P} - \mathbf{Q}^\top\mathbf{Q}

. Despite having distinct conserved quantities, both optimizers find the so-called \textit{balanced} solution from vanishing initialization. When training from small random initialization, the weights spontaneously align early in training. We derive the alignment rates in simple settings and show that they predict the empirical alignment rates in general. Finally, we exploit structural properties of Muon to construct a learning rate schedule that achieves near-perfect alignment in only two optimization steps.

Mark Rhee, Jamie Simon, Dhruva Karkada

UC Berkeley · Imbue

Muon Restricted Bernoulli Matrix Factorization Batch Matrix Gradient Balance

Jun 23, 2026math.PR

Uniform Sampling from High-dimensional Spectral Norm Balls

Motivated by an application in machine learning optimization, this paper focuses on the challenges of sampling a matrix uniformly from the unit spectral norm ball. It is proven that all singular values of sampled matrices converge to 1 almost surely as the matrix dimensions increase. This result provides the theoretical justification for a proposed simple sampling method applicable for large dimension sizes matching matrices found in modern large language models. Experimental results demonstrate both the convergence of the singular values, as well as the exact and proposed approximate sampling methods.

Michael R. Metel

Spectral Norm Singular Value Decomposition Optimal Sample Complexity Matrix Spectral High-Dimensional

Jun 16, 2026cs.LG

Non-negative Matrix Factorisation with Topological Regularisation

We investigate the learning of interpretable bases in non-negative matrix factorisation (NMF) by regularising the topology of the learned basis functions. Our approach is motivated by the observation that many data modalities can be viewed as non-negative functions on a structured domain, where the quality of a basis is intrinsically linked to its topology. However, naive methods for incorporating the topology of the support are often hindered by discreteness and threshold dependence, rendering them unsuitable for continuous optimisation. We address these challenges by employing persistent homology as a stable, threshold-free topological quantifier and by designing topological scores that integrate into the NMF objective as regularisers. The resulting framework encompasses spatially coherent image components, periodic time-series structures, and clique-like graph signals within a unified modelling language.

Matias de Jong van Lier, Shizuo Kaji, Keunsu Kim

Recursive Inc. · Tokyo, 150-0002, Japan · Graduate School of Science +5

Nonnegative Matrix Factorization Basis Functions Persistent Homology Data Manifold Matrix

Jun 13, 2026stat.ML

The Reverse Telescoping Coordinate System for Positive Definite Matrices: Geometry, Computation, and Generative Modeling

We design a new unconstrained coordinate system where a

p\times p

symmetric positive definite (SPD) matrix

Θ

is represented by a reverse telescoping map

Θ(x)=\rm{RT}(x)

, with

x=(v,d,r)\in\mathbb{R}\times\mathbb{R}^{(p-1)}\times\mathbb{R}^{p(p-1)/2}

, representing respectively the log volume or log determinant; and the shape, as encoded by log relative diagonal scales and partial covariances among the nodes. This construction results in important properties not available in other charts, e.g., matrix logarithm, such as Jacobian depending on only the log-determinant. A useful feature of our construction is

x

contains a lossless symbolic representation of both the matrix and its inverse. Many important computations involving a matrix and its inverse can be performed in

O(p^2)

in the transformed domain, while it is the rendering of results in matrix forms (on demand) that must incur an

O(p^3)

cost. Moreover, two unit-determinant matrices in the transformed domain can be joined by a straight line with pathwise unit determinant. For generative modeling, this allows designing a split volume-shape flow model trained by conditional flow matching for transporting the shape over the unit-determinant path, with a separate one-dimensional flow for transporting the volume or the determinant. The forbidding SPD constraint, tamed thus into a powerful guiding force, leads to the surprising insight that it is in some sense easier to design a volume-normalized shape flow for SPD compared to the unconstrained

\mathbb{R}^{p\times p}

, with no intrinsic notion of volume to aid normalization, unlike the determinant of SPD matrices. We apply our construction for up to

p=200

in generative modeling of SPD matrices on a difficult synthetic bimodal target, and in generating brain connectivity networks by models trained on fMRI data; as well as in intrinsic diffusion on the SPD manifold.

Anindya Bhadra

Department of Statistics, Purdue University

Matrix Generative Models Reverse Optical Flow Functional Magnetic Resonance Imaging Pathways

Jun 12, 2026cs.AR

KATANA: A Fast, Low-Power Mapping of Kalman Filters onto Edge NPUs for Real-Time Tracking

State estimation is the closed-loop core of every real-time tracking system, from radar surveillance and counter-UAV defense to autonomous driving and robotics. These deployments run on edge platforms, where defense systems mount on vehicles and drones, and civilian pipelines live on cars and handheld devices. Here, every additional watt of compute erodes mission duration or operational range. Two hard constraints follow: each new measurement must be fused before the next control cycle, and the total compute must fit within a strict battery and thermal power envelope. The Linear and Extended Kalman Filters (LKF, EKF) are dominant estimators on these systems, but today they execute almost exclusively on CPUs, which serialize multi-object tracking (MOT) updates, or on custom FPGA/ASIC accelerators that lengthen design cycles. Contemporary AI-PC SoCs, like the Intel Core Ultra Series 1 and 2, integrate a low-power, data-parallel Neural Processing Unit (NPU). We therefore ask whether the Kalman filter can be mapped onto this existing matrix engine to meet real-time and low-power budgets simultaneously, avoiding a dedicated accelerator and keeping the CPU and GPU free for primary workloads. We present KATANA, an NPU-aware optimization framework delivering the first end-to-end mapping of the LKF and EKF onto a commercial NPU, alongside a cross-platform characterization on shipping AI-PC silicon. KATANA applies three algebraic graph rewrites: subtract-to-add reformulation via a precomputed negative-projection matrix H_neg, static-shape tensor fusion, and block-diagonal batched parallelization, ensuring 100% of operations execute on the DPU matrix engine. On the Series 2, the optimized batched EKF reaches 223.35 FPS at 13.43 W active power, and the LKF reaches 408.73 FPS at 14.05 W, delivering up to a 97.9% reduction in dynamic energy versus the CPU implementation.

Bodhisatwa Kundu, Anish Rooj, Sumit Saha +4

1Jadavpur University, West Bengal, India · 2Purdue University, IN, USA · 3Intel Corporation, CA, USA

Extended Kalman Filter Neural Processing Units State Estimation Real-Time Systems Matrix

Jun 11, 2026math.NA

Approximating Gaussian Whittle-Matern Fields over Well-Centered Triangulations of Riemannian Manifolds

Markovian Whittle-Matérn fields have been convergently approximated by discrete Gauss Markov Random Fields (GMRFs) with sparse precision matrices using a Finite Element approximation of the two-parameter family,

(κ^2 - Δ)^{α/2} u = \mathcal{W}, \;\; κ\in \mathbb{R}, \; α\in \mathbb{N}.

of SPDEs. Using recent developements in the analysis of Discrete Exterior Calculus (DEC), we present a different, yet closely related, convergent GMRF approximation to these Matérn fields over complete, boundaryless Riemannian manifolds discretized as well-centered simplicial complexes. This convergent method (i) is agnostic to

α, κ

and thus allows a universal approximation scheme for the precision and covariance matrices of the entire

(α, κ)

-family of GMRFs, so they may be inferred rather than guessed. (ii) inherently models pointwise and piecewise-smoothed measurements of a random field and approximates both equally well (iii) is computationally independent of the interpolants used - it suffers no overhead if one convergent interpolant were replaced with another suitable interpolant over the same mesh. Furthermore, we show that, on discretizations that are well-connected in a precise sense, and volume-concentrated, the precision matrices are spectral functions of a graph-laplacian. We provide a low rank approximator to the family of such Matérn GMRFs and mention a use case: reducing the number of measurements needed to model the GMRF by compressed-sensing.

Srinivas Nambirajan

Riemannian Manifolds Spatiotemporal Fields Triangulations Matrix

Jun 11, 2026cs.ET

Modern analog computing for solving differential and matrix equations

In recent years, driven by the computational demands of data-intensive applications such as artificial intelligence and scientific computing, analog computing has gained renewed interest. Given the diversity of computational tasks and recent advancements in analog CMOS circuits and resistive memory technologies, we refer to the evolving landscape as modern analog computing. In this context, we identify three core computational primitives: solving differential equations, solving matrix equations, and performing matrix-vector multiplications, and we explore the connections among them. We also examine various hardware implementations of these analog computing operators, including those built with discrete components, integrated circuits, and resistive memory devices. Among these, resistive memory arrays emerge as particularly promising due to their implementation efficiency. The paper then surveys recent progress in leveraging modern analog computing to solve differential and matrix equations using both advanced analog CMOS circuits and resistive memory arrays. Finally, we discuss the applications of these circuits, the precision and scalability issues and their potential solutions, the relationship with in-memory computing, and the unique computational complexity of analog computing. This paper provides a unified perspective on analog computing, highlighting its strengths, current developments, and challenges, and positioning it as a pivotal enabler of next-generation computational frontiers.

Zhong Sun, Piergiulio Mannocci, Manuel Le Gallo +1

Institute for Artificial Intelligence, School of Integrated Circuits, Peking University, Beijing Advanced Innovation Center for Integrated Circuits, Beijing 100871, China. · 2Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milano, 20133, Italy. · 3IBM Research Europe, Rüschlikon, Switzerland.

In-Memory Computing Analog Reram Crossbar Matrix

Jun 5, 2026cs.CL

Your UnEmbedding Matrix is Secretly a Feature Lens for Text Embeddings

Large language models exhibit impressive zero-shot capabilities across a wide range of downstream tasks. However, they struggle to function as off-the-shelf embedding models, leading to suboptimal performance on massive text embedding benchmarks. In this paper, we identify a potential cause underlying this deficiency. Our motivation stems from an unexpected observation: text embeddings tend to align with frequent but uninformative tokens when projected onto the vocabulary space. We argue that this excessive expression of high-frequency tokens suppresses the model's ability to capture nuanced semantics. To address this, we introduce EmbedFilter, a simple linear transformation designed to refine text embeddings derived from LLMs directly. Specifically, we uncover that the unembedding matrix within LLMs encodes a latent space that is actively writing these frequent tokens into embedding space. By filtering out this subspace, EmbedFilter suppress the influence of high-frequency tokens, thereby enhancing semantic representations. As a compelling byproduct, this enables an inherent dimensionality reduction, lowering index storage and speedup retrieval while fully preserving the refined embedding quality. Our experiments across multiple LLM backbones demonstrate that LLMs equipped with EmbedFilter achieve superior zero-shot downstream performance even with significantly reduced embedding dimensions. We hope our findings provide deeper insights into the mechanisms of LLM-based representations and inspire more principled designs to improve text embeddings training. Our code is available at https://github.com/CentreChen/EmbFilter.

Songhao Wu, Zhongxin Chen, Yuxuan Liu +3

Gaoling School of Artificial Intelligence, Renmin University of China Beijing, China · Lenovo Group Limited Beijing, China · Wuhan University Wuhan, China

Text Embeddings Latent Space Matrix

Jun 4, 2026cs.LG

Non-Negative Matrix Factorization for Event Data

Continuous-time event data, in which entities emit instantaneous events over time, arises naturally across many domains such as neuroscience, seismology, and social networks. Non-negative matrix factorization (NMF) is a natural tool to uncover interpretable structure in such data, but it has so far only been applied after binning or smoothing the entity-level counting measures. This preprocessing step comes with the risk of erasing entity-level heterogeneities and fine-grained temporal features. In this paper, we introduce EventNMF, a continuous-time non-negative factorization model that operates directly on event times: each entity's events are modeled as a Poisson process whose intensity factorizes through a non-negative B-spline basis, and a simple estimation procedure recovers interpretable temporal templates shared across entities. The resulting method is mathematically principled, easy to implement, and computationally efficient. We further show that standard binned-count approaches arise as the special case of degree-zero splines, explore bias-variance tradeoffs and compare against existing methods on a synthetic latent factor model, and demonstrate the effectiveness of EventNMF on several real-world applications.

Raphaël Romero

Ghent University · Ghent, Belgium

Nonnegative Matrix Factorization Determinantal Point Process Exploratory Factor Analysis Matrix

May 31, 2026cs.LG

Riemannian Optimization for Hadamard Products of Low-Rank Matrices

The elementwise Hadamard product of two low-rank matrices provides a parameter-efficient model for data with multiplicative structure, but its modeling is challenging due to the presence of additional symmetries under coupled row/column scalings between the two factors. In order to leverage the geometry of the space, we formulate the learning of such matrices as optimization on a Riemannian quotient manifold. We propose a novel block-diagonal Riemannian metric derived from the pullback of the Frobenius inner product. The metric is shown to be invariant under these symmetries. We develop a Riemannian gradient descent algorithm that uses a tuning-free Gauss--Newton step size and scales linearly in the number of observed entries per iteration. The versatile framework of Riemannian quotient optimization enables both first-order and second-order Riemannian methods, the latter through a closed-form connection and the Riemannian Hessian. Experiments on real and synthetic datasets illustrate the efficacy of our proposed Riemannian approach.

Pratik Jawanpuria, Ankish Chandresh, Bamdev Mishra

Centre for Machine Intelligence and Data Science, Indian Institute of Technology Bombay, India · Microsoft India

Riemannian Optimization Low-Rank Structure Riemannian Manifolds Randomized Hadamard Transforms Matrix Matrix Multiplication

May 28, 2026stat.ML

Improved Guarantees for Heterogeneous Treatment-Effect Estimation via Matrix Completion

A central goal of modern causal inference is estimating heterogeneous treatment effects to answer questions like "how does an intervention affect each unit," rather than only on average. We study this problem with panel-data where we observe

n

units across

m

times under unknown, non-uniform treatment assignments. The data in this setting is naturally represented as a matrix of all unit--time treatment effects. Estimating heterogeneous treatment effects can then be expressed as obtaining a good estimation of each row's average in this matrix. This allows us to formulate the problem as matrix completion, which can be solved under natural low-rankness assumptions. However, existing matrix-completion guarantees are not powerful enough to get meaningful bounds for the per-row guarantee required for estimating the heterogeneous treatment effect; roughly speaking, they are only useful for estimating average treatment effect bounds, as also illustrated in a recent line of work. We give a simple, computationally efficient estimator that, without knowledge of the propensities and under standard low-rankness and regularity assumptions, achieves a row-wise

\ell_2

error of

\tilde{O}(\sqrt{\frac{1}{n} + \frac{n}{m^2}})

. Technically, our analysis establishes the first sharp row-wise

\ell_2

-perturbation bound for low-rank approximation, complementing existing spectral-, Frobenius-, and entrywise perturbation theory.

Anay Mehrotra, Phuc Tran, Van H. Vu +1

Vin University · The University of Hong Kong · Yale University

Heterogeneous Treatment Effects Tensor Completion Causal Inferences Finite-Sample Matrix

May 21, 2026cs.DC

Secure and Parallel Determinant Computation for Large-Scale Matrices in Edge Environments

The advent of edge computing has enabled resource-constrained clients to delegate intensive computational tasks to distributed edge servers, especially within Internet of Things (IoT) environments. Among such tasks, Matrix Determinant Computation (MDC) remains critical for applications in control systems, cryptography, and machine learning. However, the cubic complexity of traditional determinant algorithms makes them unsuitable for real-time processing in constrained edge scenarios. We propose a Secure Parallel Determinant Computation (SPDC) framework, which provides strong security guaranties, including privacy-preserving MDC, across N distributed edge servers. The framework achieves privacy through Composite Element Distortion (CED) - a lightweight encryption method that combines Element-wise Obfuscation (EWO) and the Panth Rotation Theorem (PRT) to conceal both structural and numerical matrix content while preserving determinant properties. Parallel LU decomposition is used to distribute encrypted matrix blocks across an arbitrary number of untrusted edge servers, enabling efficient and scalable determinant computation. A one-way communication model further reduces coordination overhead by eliminating inter-server interactions. To ensure result integrity with minimal client burden, we further introduce two verification algorithms: Q_2, a probabilistic scalar method, and Q_3, a deterministic and low-complexity alternative. Mathematical analysis demonstrates that the proposed framework provides strong privacy and security guaranties, low computational overhead, and deployment flexibility - making it well-suited for secure, scalable, and real-time MDC in distributed edge-assisted systems.

Prajwal Panth

School of Computer Engineering, KIIT Deemed to be University, Bhubaneswar, India.

Edge Devices Design Structure Matrices Matrix Obfuscation Dynamic Environments

May 18, 2026cs.RO

CosFly: Plan in the Matrix, Fly in the World

We present CosFly, a box-structured planning and multimodal simulation pipeline for aerial tracking, together with CosFly-Track, a large-scale UAV dataset for dynamic target tracking across diverse environments including urban centers, highways, rural landscapes, forests, and coastal towns. In our current implementation on CARLA, CosFly provides a modular 7-step construction pipeline that converts complex 3D worlds into structured obstacle representations for planning, then projects the resulting trajectories back into multi-modal sensor data -- including RGB images, high-precision depth maps, and semantic segmentation masks -- paired with natural language navigation instructions. A key feature is the support for configurable fixed-FOV zoom levels (one FOV setting drawn per trajectory and held constant throughout), enabling simulation of various focal lengths through camera-intrinsic adjustments. The pipeline covers the complete workflow from 3D map export through grid simplification, pedestrian and drone trajectory planning, multi-modal rendering with 6-DOF pose annotations, quality inspection, and teacher-student caption generation. We analyze two trajectory-planning paradigms for aerial target tracking: a conventional two-stage pipeline with front-end candidate generation and backend refinement, and a direct gradient-based formulation that optimizes multiple tracking constraints in a single objective. The public CosFly-Track release contains 250 validated trajectories and approximately 100,000 rendered images with complete 6-DOF drone pose annotations (position x, y, z and orientation yaw, pitch, roll). Together, the pipeline and dataset establish a scalable foundation for aerial-ground collaborative research, supporting dynamic target tracking, UAV navigation, and multi-modal perception across diverse environments.

Hanxuan Chen, Xiangyue Wang, Songsheng Cheng +8

Autel Robotics · Nanjing University · Peking University +2

Aerial Dataset World Matrix

May 18, 2026cs.LG

AMO: Operator-level Adaptive Muon Orthogonalization

Muon has recently emerged as a competitive alternative to AdamW for large-scale pre-training, with orthogonalization via Newton-Schulz (NS) iteration as its core operation. Standard Muon applies a uniform NS schedule to all parameter matrices, overlooking possible differences in orthogonalization difficulty and its impact on performance. Through a systematic empirical study, we show that this per-matrix heterogeneity is pervasive and strongly associated with matrix geometry, which evolves dynamically across operator types, training stages, and network depths. Therefore, uniform NS schedules can lead to uneven orthogonalization quality across the model. Motivated by these findings, we propose Operator-level Adaptive Muon Orthogonalization (AMO), an observe-then-commit method that measures weight geometry by operator type early in training and then uses these signals to allocate the NS budget for the remainder of training. AMO delivers consistent improvements over uniform-schedule Muon across standard, prolonged, and continual pre-training, surpassing the strongest baseline by +0.76 on Llama3.1-1.4B and +0.51 on Qwen3-1.7B in average downstream performance of 12 evaluation tasks, with gains persisting at Llama3.1-4B scale.

Xinlin Zhuang, Panyi Ouyang, Yichen Li +7

The Chinese University of Hong Kong · Shopee · MBZUAI +3

Muon Orthogonality Spectral Preconditioning Pretraining Newton Matrix

May 7, 2026cs.LG

When and Why SignSGD Outperforms SGD: A Theoretical Study Based on $\ell_1$ -norm Lower Bounds

Sign-based optimization algorithms, such as SignSGD and Muon, have garnered significant attention for their remarkable performance in training large foundation models. Despite this empirical success, we still lack a theoretical understanding of when and why these sign-based methods outperform vanilla SGD. The core obstacle is that under standard smoothness and finite variance conditions, SGD is known to be minimax optimal for finding stationary points measured by

\ell_2

-norms, thereby fundamentally precluding any complexity gains for sign-based methods in standard settings. To overcome this barrier, we analyze sign-based optimizers leveraging

\ell_1

-norm stationarity,

\ell_\infty

-smoothness, and a separable noise model, which can better capture the coordinate-wise nature of signed updates. Under this distinct problem geometry, we derive matched upper and lower bounds for SignSGD and explicitly characterize the problem class in which SignSGD provably dominates SGD. Specifically, we compare the \emph{upper bound of SignSGD} with the \emph{lower bound of SGD}, illustrating that SignSGD effectively reduces the complexity by a factor of

d

under \emph{sparse noise}, where

d

is the problem dimension. Furthermore, we elevate this framework to the matrix domain, providing an equivalent optimal lower bound for the Muon optimizer, proving that extending the sign operator to matrices preserves this optimal scaling with dimensionality. Finally, we bridge our theoretical bounds to practice, demonstrating that the theoretical superiority of SignSGD accurately predicts its faster convergence during the pretraining of a 124M parameter GPT-2 model.

Hongyi Tao, Dingzhi Yu, Lijun Zhang

State Key Laboratory of Novel Software Technology, Nanjing University, Nanjing 210023, China · School of Artificial Intelligence, Nanjing University, Nanjing 210023, China

Stochastic Gradient Descent Signsgd Spectral Norm Lower Bounds Stationarity Matrix+1

May 7, 2026cs.CV

Solving Minimal Problems Without Matrix Inversion Using FFT-Based Interpolation

Estimating camera geometry typically involves solving minimal problems formulated as systems of multivariate polynomial equations, which often pose computational challenges when using existing Gröbner-basis or resultant-based methods due to matrix inversion needed in the online solver. Here we propose a sampling-based, matrix inversion-free method that constructs the solvers using sparse hidden-variable resultants. The determinant polynomial in the hidden variable is efficiently reconstructed via inverse fast Fourier transform interpolation from sampled evaluations, avoiding symbolic expansion. Solving this polynomial yields the hidden variable, and the remaining unknowns are recovered by identifying rank-1 deficient submatrices and applying Cramer's rule. A greatest common divisor-based criterion ensures robust submatrix identification under noise. Experiments on diverse minimal problems demonstrate that the proposed solver achieves strong numerical stability and competitive runtime, particularly for small-scale problems, providing a practical alternative to traditional Gröbner-basis and resultant-based solvers.

Haidong Wu, Snehal Bhayani, Janne Heikkilä

Center for Machine Vision and Signal Analysis University of Oulu, Oulu, Finland

Solver Iterations Matrix Polynomials Interpolation