Matrix Factorization

This paper builds a theoretical bridge between the economic notion of marginal utility and two machine-learning constructs, matrix factorization and the Key--Value cache of transformer language models. The singular value spectrum of a rating matrix is shown to be a diminishing marginal utility schedule for latent factors, the eigenvalue spectrum of the projected covariance operator to be the marginal utility schedule of a model's learned representation, and cache eviction and low-rank cache compression to be instances of constrained utility maximization under a memory budget. The three collapse into a single allocation rule: retain the top dimensions whose eigenvalue exceeds the shadow price of the binding constraint. The framework is applied to the automated extraction of structured information from geo-mining documents, where it motivates a multi-pass inference protocol, a layer-wise TIES model merging procedure, and a selection policy combining extraction quality, localization drift and energy, scalarized with a Conditional Value-at-Risk term on drift. Two empirical contributions are reported. An 11.2-million-parameter hierarchical classifier, trained in about five minutes on a single GPU, reaches 90.0 per cent level-1 accuracy on a held-out test set from a 973-document uranium-exploration corpus, against 92.0 per cent for a proprietary model on a fifty-document human audit of the same corpus, at a latency of 2.62 ms per card against approximately 2,000 ms for the API and at negligible cost. A diagnostic of uniform-density TIES merging exposes a reproducible degenerate mode in which the merged model returns token-identical outputs across five geographically distinct districts while declaring high confidence; re-executing the merge under layer-wise calibrated densities removes that signature on the diagnostic sample. The full-scale extraction benchmark, including LoRA fine-tuning, is reported as projected rather than measured and remains an empirical extension of this work.

Low-rank tensor modeling has become an effective tool for hyperspectral anomaly detection. However, existing methods still suffer from high computational cost and limited flexibility in characterizing spatially structured anomalies. To address these issues, this paper proposes a hyperspectral anomaly detection method based on group sparse low-rank tensor factorization with automatic anomaly grouping (GSAA). Specifically, the low tubal rank background is characterized by imposing group sparsity on tensor factors, which provides an efficient alternative to direct tensor rank regularization. For anomaly modeling, a latent grouping map is introduced to build an automatic anomaly grouping penalty, allowing anomaly groups to be adaptively inferred from the data rather than predefined at the pixel level. To further exploit complementary spectral and spatial information, GSAA is applied in both domains, and the resulting detection maps are fused to form a spectral--spatial version of GSAA, termed GSAA-SS. An efficient linearized alternating direction method of multipliers algorithm with convergence guarantee is developed to solve the resulting model. Experimental results on five real hyperspectral datasets demonstrate that the proposed method achieves superior detection performance and competitive computational efficiency compared with several state-of-the-art methods.

Many scientific graphs attach several variables to each node, so a single scalar edge weight cannot describe direction-dependent interactions. We model each edge by a symmetric positive-definite (SPD) matrix and infer a posterior over matrix-valued graph geometries, which we call the Bayesian matrix-valued graph (BMVG). We ask how these interactions reconfigure across contexts: how large the change is and which multivariate directions strengthen or weaken. The geodesic distance induced by the affine-invariant Riemannian metric (AIRM) quantifies deformation magnitude and generalized eigenvalues resolve its signed directions.Against fused graphical lasso, Bayesian multiple-GGM, and common principal components, BMVG is competitive on global precision recovery while retaining identifiable matrix-valued edge structure and accurately recovering edge-level deformation directions. In controlled known-truth experiments, it resolves structural change with increasing sample size, including orientation changes that leave ordinary eigenvalues unchanged. In one year of Bay Area weather data, the geometry of 12-hour change reconfigures spatial coupling about as much as whole seasons differ. In TCGA-BRCA, estrogen-receptor (ER)-associated reconfiguration concentrates on specific gene-module pairs and persists under graph-scaffold sparsification and removal of subgroup mean differences. These results establish posterior matrix-valued edge geometry as a unified framework for quantifying and interpreting context-dependent multivariate reconfiguration.

In order to understand complex systems such as the human metabolome or human brain, different sensing technologies are used, generating complex data. These datasets are often multiway, i.e., with more than two axes of variation such as a subjects by metabolites by time array. While tensor factorizations have successfully revealed interpretable patterns from such complex data, they have so far been mainly data-driven. On the other hand, there is more to data -- there are computational models (of these systems), which are rich sources of prior information. In this paper, we introduce a knowledge-guided approach that brings together data and computational models by jointly analyzing real data and simulated data (generated using a computational model) using coupled tensor factorizations with linear coupling. Our experiments on real metabolomics measurements demonstrate that guiding the analysis of such noisy data with simulated data improves the pattern discovery performance while also revealing potential discrepancies between data and computational models.

Real-world recommendation platforms routinely collect explicit negative feedback such as 1-star reviews, hate-button clicks, distrust between users, and very-low watch-ratio videos. Learned sign-aware recommenders exploit this signal for clear accuracy gains, but only at the cost of gradient-based training. In parallel, a line of training-free spectral collaborative filtering methods matches or beats learned graph recommenders at a fraction of the cost, yet operates on positive interactions alone. We bridge these two lines with DualSpectralCF, a training-free framework of two components that attach to any spectral backbone of the form $\hat{\mathbf{r}}_u = F(\mathbf{M}) \mathbf{r}_u$: a signed input signal $\mathbf{r}_u^{\pm}$ that encodes the user's explicit dislikes, and a signed item-item operator $\mathbf{M}^{\pm}$ that blends like-together and dislike-together similarity. The framework is backbone-agnostic and adds just two scalar hyperparameters. We instantiate DualSpectralCF on ChebyCF, GF-CF, and Turbo-CF, and evaluate on five sign-aware benchmarks: every instance matches or beats its unsigned backbone on all 5 datasets, with Recall@20 lifts up to +32.6% with backbone-specific $(γ, κ)$ tuning and +1.9% to +16.0% for DualSpectralCF-Cheby at the fixed default $(γ= -0.5, κ= 0.1)$, and the family runs 7.7 to 155.3$\times$ faster than SIGformer while reaching 70.7% to 90.7% of its accuracy. Sign-awareness helps most for cold-start users, with up to +29.2% Recall@20 on Epinions users with 1 to 5 training items.

Matrix factorisation is a fundamental tool for exploiting low-dimensional structure in high-dimensional data, with applications such as data compression, denoising, structure discovery, interpretable representation learning, and dimensionality reduction. Compared to conventional two-factor models, matrix tri-factorisation provides greater modelling flexibility, while sparsity constraints often improve both interpretability and recovery performance. Although the role of sparsity has been extensively studied for two-factor matrix factorisation, rigorous theoretical guarantees for general real-valued matrix tri-factorisation remain largely unexplored. To address this gap, we establish, to the best of our knowledge, the first rigorous theoretical study for sparsity-induced identifiability in general real-valued matrix tri-factorisation. Our analysis is enabled by a novel decomposition strategy that transforms the original problem into two coupled auxiliary factorisation problems, while preserving the structural information necessary to the recovery of the original factor matrices from the observations. Building upon this decomposition, we derive recovery guarantees and structural consistency results that characterise how coefficient sparsity influences the sufficient recovery conditions, convergence behaviour, spectral approximation error, high-probability bounds, and structure preservation. Comprehensive Monte Carlo experiments validate the proposed theory and demonstrate close agreement between the theoretical results and empirical observations.

Singular Value Decomposition (SVD) underlies matrix factorisation tasks across many fields, with imaging applications demanding real-time processing. Yet SVD algorithms are inherently sequential, constraining real-time GPU throughput and limit online deployment in imaging pipelines. This study introduces a fully parallelized matrix factorization framework for GPUs by enforcing matrix orthogonality on left singular vectors via Lie-parametrised algebra and recovering the remaining components through soft constraints. This asymmetric constraint design enables an efficient parallel and provably valid decomposition, achieves high reconstruction fidelity and substantially accelerates computation relative to the exact SVD, with real-time throughput exceeding standard video frame rates. Performance is evaluated on multiple imaging tasks spanning complementary computational regimes: (1) spatio-temporal background subtraction for ultrasound localisation microscopy, requiring high-dimensional matrix separation, (2) Mueller matrix polarimetry for neurosurgical tissue characterisation, requiring massive batch processing of small matrices, and (3) an MNIST denoising benchmark at an intermediate scale with known ground truth. Across regimes and instruments, the proposed framework demonstrates robust domain transfer at various matrix scales, sufficient for live image-guided workflows that classical solvers cannot currently support in these settings. By prioritising downstream reconstruction fidelity over exact spectral recovery, the proposed SVD framework makes structured matrix factorisation practical for real-time processing.

Modeling shared and subject-specific structure in multisubject spatiotemporal data remains challenging, particularly in neuroimaging, where both spatial and temporal patterns exhibit rich variability across subjects. Existing matrix and tensor decompositions provide interpretable factorizations, but rely on fixed multilinear structures or coupling schemes that may limit their flexibility in capturing complex variability. In this work, we introduce a spatiotemporal variational tensor decomposition (ST-VTD) framework that combines a tensor factorization generative model with structured priors to jointly represent spatial maps and temporal dynamics. Spatial factors are regularized to promote a low-rank structure inspired by the LL1 decomposition, while temporal factors are modeled using a learned Long short-term memory (LSTM)-based prior, enabling flexible and adaptive dynamics. Posterior inference is performed using an amortized variational formulation by unrolling iterations of an optimization algorithm, leading to an interpretable and parameter-efficient architecture. The proposed inference framework employs a warm-start strategy based on group independent component analysis, which we found to improve optimization performance. Experiments on a realistic synthetic functional MRI (fMRI) dataset demonstrate that the proposed approach significantly improves latent factor recovery compared with representative classical and probabilistic decomposition benchmarks.

Non--negative matrix factorization (NMF) has become an established dimensionality reduction technique for extracting latent structures from non--negative data and has found widespread applications in fields such as bioinformatics, text mining, image analysis, and recommender systems. As the popularity of NMF has increased, numerous \textit{R} packages implementing different optimization strategies and computational frameworks have been developed. Despite their widespread availability, comprehensive evaluations of these implementations under real--world data conditions remain limited. Consequently, researchers often lack objective guidance when selecting an appropriate package for practical applications. This study introduces a new \textit{R} package for NMF and offers asystematic performance comparison with two widely available \textit{R} packages for NMF analysis. Rather than relying on simulated datasets, the evaluation is conducted using real--world data to better reflect the complexity, heterogeneity, and noise characteristics encountered in practical analytical settings. The packages are assessed using a consistent experimental framework, with emphasis on computational efficiency, convergence behavior, reconstruction accuracy, memory utilization, and the stability of the resulting matrix factorization.

In this paper, we present SCPP (Soft Clustering Python Package), an open-source Python framework for soft clustering. SCPP establishes a canonical, scikit-learn-compatible estimator interface that standardizes model training, prediction, membership representation, evaluation, and benchmarking across heterogeneous soft clustering methods, including fuzzy, probabilistic, graph-based, matrix factorization, and deep learning methods. The framework currently integrates 40 representative algorithms together with a comprehensive benchmarking comprising datasets, clustering quality metrics, and standardized runtime, memory, and scalability evaluation. SCPP further provides extensive documentation, practical examples, automated testing, and seamless integration with the scientific Python ecosystem, enabling reproducible experimentation and straightforward extension with new algorithms. The source code is publicly available at https://github.com/soft-clustering/soft-clustering.

The Matthew effect is a big challenge in Recommender Systems (RSs), where popular items tend to receive increasing attention, while less popular ones are often overlooked, perpetuating existing disparities. Although many existing methods attempt to mitigate Matthew effect in the static or quasi-static recommendation scenarios, such issue will be more pronounced as users engage with the system over time. To this end, we propose a novel framework, Multi-Hypergraph Boosted Multi-Interest Self-Supervised Learning for Conversational Recommendation (HiCore), aiming to address Matthew effect in the Conversational Recommender System (CRS) involving the dynamic user-system feedback loop. It devotes to learn multi-level user interests by building a set of hypergraphs (i.e., item-, entity-, word-oriented multiple-channel hypergraphs) to alleviate the Matthew effec. Extensive experiments on four CRS-based datasets showcase that HiCore attains a new state-of-the-art performance, underscoring its superiority in mitigating the Matthew effect effectively. Our code is available at https://github.com/zysensmile/HiCore.

A common method for the representation and analysis of time-series data is the hidden Markov model (HMM), where each observation is associated with a hidden state that evolves over time. However, many real-world systems are influenced by multiple independent factors, which are more naturally represented by factorial hidden Markov models (fHMM), where several hidden Markov chains jointly generate the observed data. Although an fHMM provides a richer and more realistic representation of many real-world systems, it can be reformulated as an equivalent HMM, but with a significantly larger state-space, leading to a severe increase in computational cost. In particular, the forward filtering algorithm, which is central to evaluation, decoding, and estimation tasks, becomes prohibitively expensive even for small systems. This work focuses on developing scalable methods for time-series analysis using tensor algebra to exploit the multidimensional structure of fHMM directly, without constructing intermediate HMM representations. Our novel filtering approach significantly improves computational performance and enables the efficient analysis of large systems and datasets, extending the scope of fHMM and providing a practical framework for data intensive applications.

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.

Binary data factorization is common, but real-valued methods ignore discreteness and yield hard-to-interpret factors. Boolean Matrix Factorization (BooMF) instead decomposes a binary matrix into two lower-rank binary matrices via logical AND and OR, expressing the data as a Boolean disjunction of interpretable patterns. In cancer genomics, BooMF can reveal coordinated feature changes that may drive tumor evolution, unlike rotational or additive decompositions. Most existing BooMF methods are heuristic, greedy, sensitive to initialization, prone to local optima, and do not support principled model selection or uncertainty quantification. We introduce Bayesian Boolean Matrix Factorization (BBMF), a fully conjugate generative model with sparsity-inducing priors. It enforces Boolean constraints, yields interpretable latent factors with coherent uncertainty quantification, and admits Gibbs sampling with closed-form full conditionals. Because cancer evolution often involves widespread, near-simultaneous chromosome-number changes (e.g., whole-genome duplication followed by instability and selection), Boolean factorizations capture these patterns more naturally than additive models. Applied to arm-level copy-number alteration data in multiple myeloma, where entries indicate presence/absence of chromosomal-arm amplifications, BBMF finds a small set of interpretable bicliques linking patient subsets to recurrently co-altered chromosomal arms, providing a compact, biologically meaningful summary of tumor heterogeneity and demonstrating BBMF's utility for uncovering discrete latent structure in complex binary data.

Incorporating textual reviews into a Recommender System has become a prominent strategy for enriching collaborative signals with semantic information. However, the actual contribution of review-derived representations remains an open question, particularly when strong collaborative baselines are employed. In this work, we systematically investigate the impact of textual information on Matrix Factorization by introducing and comparing three enrichment strategies over a common collaborative backbone. First, we propose a learnable gating mechanism that adaptively balances collaborative and textual signals during training. This mechanism is applied to two distinct review representations: (i) aggregated topic profiles extracted from user and item histories, and (ii) full text embedding representations derived from reviews. Additionally, we explore a cross-attention mechanism that identifies and emphasizes the most informative dimensions of the textual representation before fusion with collaborative factors. We evaluate six variants: pure, enriched with topic profiles and text via gating; enriched with topics and text via gating; and enhanced with cross-attention over textual features. Experiments across multiple review-based datasets reveal that although adaptive fusion mechanisms improve representation flexibility, the marginal contribution of textual signals remains limited compared to the collaborative backbone. These findings suggest that, under typical rating-prediction settings, collaborative information continues to dominate performance, raising important considerations for the effective integration of semantic review signals into recommendation models.

Matrix completion has been extensively studied for real-valued data, but existing methods are often limited in handling categorical variables. We propose LCMC, a double-loop optimization framework for categorical matrix completion via latent factorization based on a binary tensor representation. In this setting, each categorical entry is encoded as a one-hot vector along a third tensor mode, thereby preserving its discrete, non-ordinal nature. The outer loop adaptively estimates the latent dimension by iteratively updating it with feedback from the inner loop, while the inner loop reconstructs the categorical matrix through tensor factorization, supported by a corresponding theoretical analysis. To further improve scalability and robustness, we introduce enhancements including a split-merge-refine strategy and an adaptive data reduction technique. Experiments on synthetic and real-world datasets in viral quasispecies reconstruction, demonstrate that LCMC achieves superior accuracy and efficiency compared to existing methods.

Community-based fact-checking that relies on cross-consensus is expanding rapidly on social media platforms. However, the delay and low-ratio of cross-consensus community fact-checks rated by human contributors remains a significant challenge. To address this, we first created ComRate, a large-scale dataset comprising 2.5 million community notes and over 209 million ratings sourced from $\mathbb{X}$. We then propose MultiCom, a persona-guided multi-agent rating framework for community note evaluation. MultiCom simulates diverse rater population by clustering contributors in a matrix-factorized rater space and prompting persona agents to generate structured assessments based on the official community notes rating schema. These agents output structured and explainable judgments, such as confidence, agreement signals and reasons. An out-of-fold calibrated aggregation algorithm combines features such as raw votes and diagnostic reason signals for reliable prediction. Extensive evaluations demonstrate that MultiCom outperforms alternative methods, achieving an average accuracy of 84.7% (balanced accuracy 68.3%, macro-F1 60.1%) on the evaluation set.

Non-negative reduced biquaternion matrix factorization (NRBMF) uses the product of reduced biquaternion (RB) matrices to incorporate the non-negativity constraints of color image pixels into the factorization process. However, NRBMF mainly focuses on reconstruction accuracy and does not explicitly exploit the local geometric structure of image data, which may limit the discriminative ability of the obtained low-dimensional coefficient representations. To address this issue, we propose a graph regularized non-negative reduced biquaternion matrix factorization (GNRBMF) model for color image recognition. The proposed model incorporates a graph Laplacian regularizer into the reduced biquaternion coefficient matrix, encouraging nearby samples in the original space to have similar coefficient representations. Meanwhile, GNRBMF retains the non-negativity property of NRBMF in the reduced biquaternion algebra. To solve the optimization problem, a component-wise alternating projected gradient algorithm is derived, and its convergence properties are analyzed. Experimental results on three color image datasets show that the proposed GNRBMF model achieves competitive or superior recognition performance compared with several methods in most tested settings.

We explore Riemannian optimization techniques for rank-factored matrix parameters, targeting contemporary deep learning applications. We examine ten points in the algorithm design space: two geometries for rank-$r$ matrices, three geometries for rank-$r$ partial isometries, and block-matrix variants of these five, where factors are shared across block-rows and block-columns. We apply our methods to the multihead attention parameters in small language models. After tuning learning rates, our methods do not conclusively outperform an AdamW baseline. Our implementations are available online.

This paper studies the stability of low-rank implicit regularization in perturbed deep matrix factorization, where the target matrix is corrupted by a noise matrix. We first derive sufficient spectral conditions under which gradient descent exhibits a low-rank phase in the noiseless setting. These conditions show how the target spectrum, initialization, and step size jointly determine the existence of a nonempty low-rank interval. We then analyze the perturbed gradient descent dynamics, proving convergence guarantees and quantifying how the perturbation affects iteration complexity and eigenvalue recovery. Finally, we show that the low-rank phase persists under perturbation, with explicit dependence on the perturbation size. Numerical experiments support the theoretical findings.

Inferring time-varying graph structures from high-dimensional nodal observations is a fundamental problem arising in neuroscience, finance, climatology, and beyond. Two intrinsic challenges govern this problem: maintaining the \emph{temporal coherence} of the latent graph across successive observation windows, and respecting the \emph{intrinsic Riemannian geometry} of the symmetric positive definite manifold on which precision matrices naturally reside, a curved space whose geodesic structure departs fundamentally from that of the ambient Euclidean space. In this paper we propose dynamic estimation on the Grassmann manifold with a factor model (\textsc{Degfm}), a novel algorithm that jointly addresses both challenges. We model the time-varying precision matrix sequence as a low-rank-plus-diagonal structure governed by a latent elliptical graph factor model, which drastically reduces the effective parameter count and enables reliable estimation in the challenging small-sample regime. Temporal coherence is enforced through a Riemannian geodesic penalty defined on the Grassmann manifold, ensuring that the estimated graph trajectory is smooth with respect to the intrinsic geometry rather than the ambient Euclidean space. To solve the resulting non-convex optimization problem over Grassmann-manifold-valued sequences subject to the LRaD constraint, we derive an efficient Riemannian gradient descent algorithm that respects the manifold structure at every iterate and rigorously establish its convergence to a stationary point. Extensive experiments on both synthetic benchmarks and real-world datasets demonstrate that \textsc{Degfm} consistently outperforms state-of-the-art baselines across all evaluation metrics, confirming the practical effectiveness of the proposed framework.

Rearranging the rows or columns of a sparse matrix using an appropriate ordering can significantly reduce fill-ins, i.e., new nonzeros introduced during matrix factorization, decreasing memory usage and runtime. However, finding an ordering that minimizes fill-ins is NP-complete. Existing approaches, including graph-theoretic and deep learning methods, rely on surrogate objectives without theoretical guarantees. The Fill-Path Theorem reveals a direct and intrinsic relationship between fill-in generation and the sparse structure of the matrix as path triplet inequalities. Here we first employ a multigrid graph network to capture structural information for each vertex. We then derive a triplet sampling strategy based on inequalities. Finally, we introduce an end-max chain loss function to reduce the number of triplets whose predicted scores satisfy these inequalities. Experimental evaluations on the publicly available SuiteSparse matrix collection demonstrate the superiority of the proposed method in terms of both fill-in reduction and speedup in LU factorization time.

Sparse matrix reordering can significantly reduce the fill-in during matrix factorization, thereby decreasing the computational and storage requirements in sparse matrix computations. Finding a minimal fill-in ordering is known to be an NP-hard problem. Moreover, there is a paradox: matrix reordering is applied before matrix factorization, but fill-ins that matrix reordering methods aim at are generated from matrix factorization. To bridge the gap between reordering and factorization, we propose a deep learning framework to minimize a fill-in surrogate function based on spectral embedding. First, we employ a multi-grid-like GNN architecture to learn to approximate the smallest eigenvectors of its graph Laplacian matrix, i.e. spectral embedding, and capture the global structural information of the matrix. Then, another multi-grid-like GNN architecture is used to minimize the potential space where fill-in can occur based on the rank distribution. Experimental results indicate that our approach achieves competitive performance compared with traditional graph-theoretic algorithms and deep learning methods.

Human trajectory forecasting is crucial for safe navigation in crowded environments, requiring models that balance accuracy with computational efficiency. Efficiently modeling social interactions is key to performance in dense crowds. Yet, most recent methods rely on attention mechanisms, which are effective at capturing complex dependencies, but incur quadratic computational costs that scale poorly with the growing number of neighbors. Recently, Selective State-Space Models have provided a linear-time alternative; however, their inherently sequential design is misaligned with the unstructured and dynamic nature of social interactions. To address this challenge, we propose Social-Mamba, a forecasting architecture that reformulates social interactions as structured sequential processes. At its core is the Cycle Mamba block, a novel module that enables continuous bidirectional information flow. Social-Mamba organizes agents on an egocentric grid and introduces social triplet factorization, which decomposes interactions into temporal, egocentric, and goal-centric scans. These are dynamically integrated through a learnable social gate and global scan to generate accurate and efficient trajectory predictions. Extensive experiments on five trajectory forecasting benchmarks show that Social-Mamba achieves state-of-the-art accuracy while offering superior parameter efficiency and computational scalability. Furthermore, embedding Social-Mamba into a flow-matching framework further enhances both accuracy and efficiency, establishing it as a flexible and robust foundation for future trajectory forecasting research. The code is publicly available: https://github.com/vita-epfl/Social-Mamba

We present Supervised Deep Multimodal Matrix Factorization (SD3MF), an interpretable framework for integrative brain network analysis that generalizes Symmetric Nonnegative Matrix Tri-Factorization (SNMTF) from unsupervised single-graph clustering to supervised prediction over populations of multimodal graphs. SD3MF learns deep hierarchical factorizations for each modality together with a shared latent representation that aligns subjects across views. An encoder-decoder formulation jointly optimizes graph reconstruction and supervised prediction, while adaptive weights enable data-driven multimodal fusion. By representing each subject through community-level interaction matrices, the model yields interpretable and discriminative features. Experiments on multimodal connectome datasets show that SD3MF consistently outperforms strong deep learning baselines such as CNNs and GNNs, while enabling biologically interpretable insights. Code for reproducibility is available at: https://github.com/amjadseyedi/SD3MF.

In this paper, we propose DiBA (Diagonal and Binary Matrix Approximation), a compact matrix factorization for neural network weight compression. Many components of modern networks, including linear layers, $1\times1$ convolutions, attention projections, and embedding layers, have dense matrix weights. DiBA approximates $A\in\mathbb{R}^{m\times n}$ by $\widehat A=D_1B_1D_2B_2D_3$, where $D_1,D_2,D_3$ are diagonal matrices and $B_1,B_2$ are $0/1$ binary matrices. The intermediate dimension $k$ controls the trade-off between theoretical storage and approximation accuracy. For matrix-vector products, DiBA decomposes dense multiplication into three element-wise scaling operations and two binary mixing operations, reducing the floating-point multiplication count from $mn$ to $m+k+n$. For optimization, we introduce DiBA-Greedy, an alternating solver that combines closed-form least-squares updates for the diagonal factors with exact one-bit improvement tests for the binary factors. We also introduce DiBARD (DiBA with Retuning only Diagonal factors), which replaces dense-matrix layers by DiBA factors, freezes the binary matrices, and retunes only the diagonal entries on downstream data. This preserves compact binary mixing without discrete search during adaptation. On 40 dense weight matrices extracted from public pretrained models, DiBA-Greedy yields consistent SNR improvements as the theoretical storage ratio increases. After DiBA replacement in two component-replacement studies, DiBARD improves DistilBERT/WikiText masked-token accuracy from 0.4447 to 0.5210 and Speech Commands test accuracy for an Audio Spectrogram Transformer from 0.7684 to 0.9781 without reoptimizing the binary factors.

Multi-task learning (MTL) has emerged as a pivotal paradigm in machine learning by leveraging shared structures across multiple related tasks. Despite its empirical success, the development of likelihood-based efficiently solvable algorithms--even for shared linear representations--remains largely underdeveloped, primarily due to the non-convex structure intrinsic to matrix factorization. This paper introduces a first-order algorithm that jointly learns a shared representation and task-specific parameters, with guaranteed efficiency. Notably, it converges in $\widetilde{\mathcal{O}}(1)$ iterations and attains a \emph{near-optimal} estimation error of $\widetilde{\mathcal{O}}(dk/(TN))$, \emph{improving} over existing likelihood-based methods by a factor of $k$, where $d$, $k$, $T$, $N$ denote input dimension, representation dimension, task count, and samples per task, respectively. Our results justify that likelihood-based first-order methods can efficiently solve the MTL problem.

Motivated by sensing modalities in modern autonomous systems that involve hardware-constrained spatial sampling over large arrays with limited coherence time, we develop a novel framework for rapid super-resolution multi-signal direction-of-arrival (DoA) estimation based on Hankel-structured sensing and data matrix decomposition of arbitrary rank, under both the $L_2$ and $L_1$-norm formulation. The resulting $L_2$-norm estimator is shown to be maximum-likelihood optimal in white Gaussian noise. The $L_1$-norm estimator is shown to be maximum-likelihood optimal in independent, identically distributed (i.i.d.) isotropic Laplace noise, offering broad robustness to impulsive interference and corrupted measurements commonly encountered in practice. Extensive simulations demonstrate that the proposed methods exhibit powerful super-resolution capabilities, requiring significantly lower SNR and achieving substantially higher resolution probability than recent competing approaches.

With the increasing availability of online information, recommender systems have become an important tool for many web-based systems. Due to the continuous aspect of recommendation environments, these systems increasingly rely on contextual multi-armed bandits (CMAB) to deliver personalized and real-time suggestions. A critical yet underexplored component in these systems is the representation of user state, which typically encapsulates the user's interaction history and is deeply correlated with the model's decisions and learning. In this paper, we investigate the impact of different embedding-based state representations derived from matrix factorization models on the performance of traditional CMAB algorithms. Our large-scale experiments reveal that variations in state representation can lead to improvements greater than those achieved by changing the bandit algorithm itself. Furthermore, no single embedding or aggregation strategy consistently dominates across datasets, underscoring the need for domain-specific evaluation. These results expose a substantial gap in the literature and emphasize that advancing bandit-based recommender systems requires a holistic approach that prioritizes embedding quality and state construction alongside algorithmic innovation. The source code for our experiments is publicly available on https://github.com/UFSCar-LaSID/bandits_blind_spot.

We study gradient descent for rank-1 matrix factorization through a state-dependent Lyapunov perspective. The central object is a parameterized quadratic certificate $I(δ;\,\cdot)$ whose boundary-inward property induces a monotone state parameter $δ_t$, thereby certifying that the trajectory is confined to a shrinking family of level sets. For certified initializations below the critical step size, this mechanism proves convergence to global minimizers. Above the critical step size, the same monotone-state mechanism instead leads to a balanced terminal regime; for a range of post-critical step sizes, the reduced dynamics exhibit period-2 behavior consistent with edge-of-stability phenomena. We further show that the scalar certificate is not an ad hoc algebraic construction: under structural axioms and a natural state-parameter normalization, it is uniquely determined by the monotonicity mechanism. Numerical experiments suggest that this state-dependent Lyapunov mechanism persists beyond the proved cases, including two-dimensional rank-1 approximation and quartic augmentations of scalar factorization.

Kolmogorov-Arnold Networks (KANs) are a recent neural network architecture offering an alternative to Multilayer Perceptrons (MLPs) with improved explainability and expressibility. However, KANs are significantly slower than MLPs due to the recursive nature of B-spline function computations, limiting their application. This work addresses these issues by proposing a novel base-spline Linear-Time B-splines Kolmogorov-Arnold Network (LTBs-KAN) with linear complexity. Unlike previous methods that rely on the Boor-Mansfield-Cox spline algorithm or other computationally intensive mathematical functions, our approach significantly reduces the computational burden. Additionally, we further reduce model's parameter through product-of-sums matrix factorization in the forward pass without sacrificing performance. Experiments on MNIST, Fashion-MNIST and CIFAR-10 demonstrate that LTBs-KAN achieves good time complexity and parameter reduction, when used as building architectural blocks, compared to other KAN implementations.

We prove a centre manifold theorem for a map along a manifold-with-boundary of fixed points, and provide an application to the study of gradient descent with large step size on two-layer matrix factorisation problems.

In recent years, neural networks and other complex models have dominated recommender systems, often setting new benchmarks for state-of-the-art performance. Yet, despite these advancements, award-winning research has demonstrated that traditional matrix factorization methods can remain competitive, offering simplicity and reduced computational overhead. Hybrid models, which combine matrix factorization with newer techniques, are increasingly employed to harness the strengths of multiple approaches. This paper proposes a novel ensemble method that unifies user-item and item-item recommendations through a weighted similarity framework to deliver top-N recommendations. Our approach is distinctive in its use of shared user and item embeddings for both recommendation strategies, simplifying the architecture and enhancing computational efficiency. Extensive experiments across multiple datasets show that our method achieves competitive performance and is robust in varying scenarios that favor either user-item or item-item recommendations. Additionally, by eliminating the need for embedding-specific fine-tuning, our model allows for the seamless reuse of hyperparameters from the base algorithm without sacrificing performance. This results in a method that is both efficient and easy to implement. Our open-source implementation is available at https://github.com/UFSCar-LaSID/weighted-sims-recommender.

Many robotic tasks, such as inverse kinematics, motion planning, and contact-rich manipulation, can be formulated as optimization problems. Solving these problems requires addressing inherent nonlinear kinematics, complex contact dynamics, long-horizon correlations, and multi-modal optimization landscapes, each posing distinct challenges for state-of-the-art optimizers. While existing methods tackle these issues through problem-specific strategies, such specialization inherently limits cross-task generalization, requires heavy engineering effort in problem reformulation, and hinders multi-task autonomy. Monte Carlo Tree Search (MCTS) offers a compelling framework that generalizes across diverse robotic tasks via strategic exploration of the solution space. However, it typically suffers from combinatorial complexity when applied naively, resulting in slow convergence and excessive storage space in high-dimensional domains. To address this limitation, we propose Tensor Train Tree Search (TTTS), which leverages tensor factorization to exploit implicit correlations among different branches within the decision tree. By utilizing the resulting compact, linear-complexity representation, TTTS significantly reduces both computation and storage overhead, thereby enabling highly efficient global decision making. Experimental results across inverse kinematics, motion planning around obstacles, legged robot manipulation, multi-stage motion planning, and bimanual whole-body manipulation demonstrate the efficiency of TTTS for generalized robot optimization over a diverse set of tasks.

This study investigates the robustness of shallow graph embedding methods for community detection in the face of network perturbations, specifically node deletions. Graph embedding techniques, which represent nodes as low-dimensional vectors, are widely used for various graph machine learning tasks due to their ability to capture structural properties of networks effectively. However, the impact of perturbations on the performance of these methods remains relatively understudied. The research considers state-of-the-art shallow graph embedding methods from two families: matrix factorization (e.g., LE, LLE, HOPE, M-NMF) and random walk-based (e.g., DeepWalk, LINE, node2vec). Through experiments conducted on both synthetic and real-world networks, the study reveals varying degrees of robustness within each family of shallow graph embedding methods. The robustness is found to be influenced by factors such as network size, initial community partition strength, and the type of perturbation. Notably, node2vec and LLE consistently demonstrate higher robustness for community detection across different scenarios, including networks with degree and community size heterogeneity. These findings highlight the importance of selecting an appropriate shallow graph embedding method based on the specific characteristics of the network and the task at hand, particularly in scenarios where robustness to perturbations is crucial.

Reliability measures associated with the prediction of the machine learning models are critical to strengthening user confidence in artificial intelligence. Therefore, those models that provide not only predictions, but also reliability, enjoy greater popularity. In the field of recommender systems, reliability is crucial, since users tend to prefer those recommendations that are sure to interest them, that is, high predictions with high reliabilities. In this paper, we propose Restricted Bernoulli Matrix Factorization (ResBeMF), a new algorithm aimed at enhancing the performance of classification-based collaborative filtering. This model is based on a collection of restricted matrix factorizations that jointly generate, for each user-item pair, a full probability distribution over the possible rating scores. To prove its effectiveness, the proposed model has been compared to other existing solutions in the literature in terms of prediction quality (Mean Absolute Error and accuracy scores), prediction quantity (coverage score) and recommendation quality (Mean Average Precision score). The experimental results demonstrate that the proposed model provides a good balance in terms of the quality measures used compared to other recommendation models.