arXiv subjects

Nicolas Gillis

Publications and source records attributed to Nicolas Gillis.

At least 19 recordsLinked to original sources

arXiv2026

On the Complexity of Low-Rank Matrix Signing and Entrywise Power Matrix Factorization

Given a nonnegative matrix $X$, a factorization rank $r$ and {a positive integer $p$}, entrywise power matrix factorization (EPMF) looks for a low-rank matrix $X_r$ such that $X = |X_r|^{\circ p}$ (exact case) or $X \approx |X_r|^{\circ p}$ (approximate case), where $(\cdot)^{\circ p}$ denotes the componentwise exponent. EPMF includes the modulus model ($p=1$) and componentwise square factorization ($p=2$) as special cases, the latter being closely related to the square root rank. We analyze the computational complexity of the exact decision problem and the Frobenius-norm approximation problem, and establish a complete complexity landscape. In the exact case, we show that EPMF is equivalent to the combinatorial problem of flipping the signs of the entries of a given matrix $X$ to obtain a rank-$r$ matrix, which we refer to as the low-rank matrix signing (LRMS) problem. We first show that LRMS, and hence exact EPMF, is strongly NP-hard, improving a weak NP-hardness result for the square-root-rank (Math. Prog., 2015). We then show that LRMS can be solved in polynomial time when $r$ is fixed. Moreover, when the rank $r$ is part of the input, we show that for generic matrices the algorithm is fixed-parameter tractable (FPT) in the parameter $r$; in fact, the running time is fixed-parameter linear in the number of entries of the input matrix. In the approximate case using the Frobenius norm as an error measure, we show that EPMF is NP-hard, already when $r=2$, the smallest nontrivial case.

cs.CC↗

arXiv2026

Manifold-based Algorithms for the Hadamard Decomposition

Given a matrix $X$, and two ranks $r_1$ and $r_2$, the Hadamard decomposition (HD) looks for two low-rank matrices, $X_1$ of rank $r_1$ and $X_2$ of rank $r_2$, both of the same size as $X$, such that $X\approx X_1\circ X_2$, where $\circ$ is the Hadamard (element-wise) product. In most cases, HD is more expressive than standard low-rank approximations such as the truncated singular value decomposition (TSVD), as it can represent higher-rank matrices with the same number of parameters; this is because the rank of $X_1 \circ X_2$ is generically equal to $r_1 r_2$. In this paper, we first present some theoretical insights for HD, in particular a useful reformulation $X\approx WH^\top$ where $W$ and $H$ have $r_1 r_2$ columns and belong to certain manifolds. These allow us to develop three new algorithms for computing HD. The first one uses the representation $X\approx X_1\circ X_2$ and relies on the Manopt toolbox. The other two rely on the reformulation $X\approx WH^\top$: one is a block projected gradient method, and the other is a manifold-based gradient descent algorithm that does not require projection onto the feasible set. The last two algorithms are particularly effective for handling large sparse data. We also propose new initializations that allow us to improve the accuracy of the HD. We compare our algorithms and initialization strategies with the TSVD and with the state of the art. Numerical results show that the new methods are efficient and competitive on both synthetic and real data.

math.OC↗

arXiv2026

Supervised Deep Multimodal Matrix Factorization for Interpretable Brain Network Analysis

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.

cs.LG↗

arXiv2026

Computing Lower Bounds on the Nonnegative Rank via Non-Convex Optimization Solvers

The nonnegative rank of a nonnegative matrix $X$ is the smallest number of nonnegative rank-one factors that sum to $X$. Since computing the nonnegative rank is NP-hard, it is common to circumvent this issue by computing lower and upper bounds. In this paper, we propose non-convex formulations and practical implementations for four important lower bounds for the nonnegative rank, namely the fooling set bound (FSB), the rectangle covering bound (RCB), the hyperplane separation bound (HSB), and the self-scaled bound (SSB). In particular, our algorithm for computing the SSB is the first available in the literature, to the best of our knowledge. It allows us to improve the best known lower bound on the nonnegative rank for some matrices. In some cases, they coincide with the best known upper bound, thereby establishing their exact nonnegative rank for the first time. Moreover, on canonical benchmarks, we show that our non-convex approaches provide a meaningful and often competitive alternative to standard methods. The paper also provides a consolidated reference for the current state of several classical lower bounds on a large number of benchmark matrices.

math.OC↗

arXiv2026

Nonnegative Matrix Factorization in the Component-Wise L1 Norm for Sparse Data

Nonnegative matrix factorization (NMF) approximates a nonnegative matrix, $X$, by the product of two nonnegative factors, $WH$, where $W$ has $r$ columns and $H$ has $r$ rows. In this paper, we consider NMF using the component-wise L1 norm as the error measure (L1-NMF), which is suited for data corrupted by heavy-tailed noise, such as Laplace noise or salt and pepper noise, or in the presence of outliers. Our first contribution is an NP-hardness proof for L1-NMF, even when $r=1$, in contrast to the standard NMF that uses least squares. Our second contribution is to show that L1-NMF strongly enforces sparsity in the factors for sparse input matrices, thereby favoring interpretability. However, if the data is affected by false zeros, too sparse solutions might degrade the model. Our third contribution is a new, more general, L1-NMF model for sparse data, dubbed weighted L1-NMF (wL1-NMF), where the sparsity of the factorization is controlled by adding a penalization parameter to the entries of $WH$ associated with zeros in the data. The fourth contribution is a new coordinate descent (CD) approach for wL1-NMF, denoted as sparse CD (sCD), where each subproblem is solved by a weighted median algorithm. To the best of our knowledge, sCD is the first algorithm for L1-NMF whose complexity scales with the number of nonzero entries in the data, making it efficient in handling large-scale, sparse data. We perform extensive numerical experiments on synthetic and real-world data to show the effectiveness of our new proposed model (wL1-NMF) and algorithm (sCD).

cs.LG↗

arXiv2026

Computing the Nonnegative Low-Rank Leading Eigenmatrix and its Applications to Markov Grids and Metzler Operators

We consider in this paper the problem of computing a nonnegative low-rank approximation of the rightmost eigenpair of a linear matrix-valued real operator. We propose an algorithm based on the time integration of a suitable differential system, whose solution is parametrized according to a nonnegative factorization. The conservation of the nonnegativity is theoretically motivated by the Perron-Frobenius theorem, while the computation of the rightmost eigenpair is motivated by two applications: (1) a new class of Markov chains, which we called Markov grids, whose transition matrices can be decomposed as the sum of Kronecker products, and (2) spatially structured systems in growth-diffusion operators arising for example in population and epidemic dynamics. Theoretical analysis and computational experiments show the effectiveness of the algorithm compared to standard approaches.

math.NA↗

arXiv2026

Maximum-Volume Nonnegative Matrix Factorization

Nonnegative matrix factorization (NMF) is a popular data embedding technique. Given a nonnegative data matrix $X$, it aims at finding two lower dimensional matrices, $W$ and $H$, such that $X\approx WH$, where the factors $W$ and $H$ are constrained to be element-wise nonnegative. The factor $W$ serves as a basis for the columns of $X$. In order to obtain more interpretable and unique solutions, minimum-volume NMF (MinVol NMF) minimizes the volume of $W$. In this paper, we consider the dual approach, where the volume of $H$ is maximized instead; this is referred to as maximum-volume NMF (MaxVol NMF). MaxVol NMF is identifiable under the same conditions as MinVol NMF in the noiseless case, but it behaves rather differently in the presence of noise. In practice, MaxVol NMF is much more effective to extract a sparse decomposition and does not generate rank-deficient solutions. In fact, we prove that the solutions of MaxVol NMF with the largest volume correspond to clustering the columns of $X$ in disjoint clusters, while the solutions of MinVol NMF with smallest volume are rank deficient. We propose two algorithms to solve MaxVol NMF. We also present a normalized variant of MaxVol NMF that exhibits better performance than MinVol NMF and MaxVol NMF, and can be interpreted as a continuum between standard NMF and orthogonal NMF. We illustrate our results in the context of hyperspectral unmixing. The code is available from https://gitlab.com/vuthanho/maxvolmf.jl.

cs.LG↗

arXiv2026

Matrix Factorization Framework for Community Detection under the Degree-Corrected Block Model

Community detection is a fundamental task in data analysis, and block models provide an approach for identifying a wide variety of community structures while offering high interpretability. The degree-corrected block model (DCBM) is an established model that accounts for the heterogeneity of node degrees. However, inference methods are computationally costly and highly sensitive to initialization, while cheaper alternatives, such as spectral or modularity-based approaches, are restricted to detecting specific structures, typically assortative. In this work, we show that DCBM inference can be reformulated as a constrained nonnegative matrix factorization problem. Leveraging this insight, we propose a novel method for community detection and a theoretically well-grounded initialization strategy that provides an initial estimate of communities for inference algorithms. Our approach is agnostic to any specific network structure and applies to graphs with any structure representable by a DCBM. Experiments on synthetic and real benchmark networks show that our method detects communities comparable to those found by DCBM inference while being faster; for instance, it processes a graph with 100,000 nodes and 1,000,000 edges in approximately 4 minutes. Moreover, the proposed initialization strategy significantly improves solution quality and reduces the number of iterations required by all tested inference algorithms. Overall, this work provides a scalable and robust framework for community detection and highlights the benefits of a matrix-factorization perspective for the DCBM.

cs.SI↗

arXiv2025

Alternating Direction Method of Multipliers for Nonlinear Matrix Decompositions

We present an algorithm based on the alternating direction method of multipliers (ADMM) for solving nonlinear matrix decompositions (NMD). Given an input matrix $X \in \mathbb{R}^{m \times n}$ and a factorization rank $r \ll \min(m, n)$, NMD seeks matrices $W \in \mathbb{R}^{m \times r}$ and $H \in \mathbb{R}^{r \times n}$ such that $X \approx f(WH)$, where $f$ is an element-wise nonlinear function. We evaluate our method on several representative nonlinear models: the rectified linear unit activation $f(x) = \max(0, x)$, suitable for nonnegative sparse data approximation, the component-wise square $f(x) = x^2$, applicable to probabilistic circuit representation, and the MinMax transform $f(x) = \min(b, \max(a, x))$, relevant for recommender systems. The proposed framework flexibly supports diverse loss functions, including least squares, $\ell_1$ norm, and the Kullback-Leibler divergence, and can be readily extended to other nonlinearities and metrics. We illustrate the applicability, efficiency, and adaptability of the approach on real-world datasets, highlighting its potential for a broad range of applications.

eess.SP↗

arXiv2025

Algorithms for Boolean Matrix Factorization using Integer Programming and Heuristics

Boolean matrix factorization (BMF) approximates a given binary input matrix as the product of two smaller binary factors. Unlike binary matrix factorization based on standard arithmetic, BMF employs the Boolean OR and AND operations for the matrix product, which improves interpretability and reduces the approximation error. It is also used in role mining and computer vision. In this paper, we first propose algorithms for BMF that perform alternating optimization (AO) of the factor matrices, where each subproblem is solved via integer programming (IP). We then design different approaches to further enhance AO-based algorithms by selecting an optimal subset of rank-one factors from multiple runs. To address the scalability limits of IP-based methods, we introduce new greedy and local-search heuristics. We also construct a new C++ data structure for Boolean vectors and matrices that is significantly faster than existing ones and is of independent interest, allowing our heuristics to scale to large datasets. We illustrate the performance of all our proposed methods and compare them with the state of the art on various real datasets, both with and without missing data, including applications in topic modeling and imaging.

cs.IR↗

arXiv2025

A Provably-Correct and Robust Convex Model for Smooth Separable NMF

Nonnegative matrix factorization (NMF) is a linear dimensionality reduction technique for nonnegative data, with applications such as hyperspectral unmixing and topic modeling. NMF is a difficult problem in general (NP-hard), and its solutions are typically not unique. To address these two issues, additional constraints or assumptions are often used. In particular, separability assumes that the basis vectors in the NMF are equal to some columns of the input matrix. In that case, the problem is referred to as separable NMF (SNMF) and can be solved in polynomial-time with robustness guarantees, while identifying a unique solution. However, in real-world scenarios, due to noise or variability, multiple data points may lie near the basis vectors, which SNMF does not leverage. In this work, we rely on the smooth separability assumption, which assumes that each basis vector is close to multiple data points. We explore the properties of the corresponding problem, referred to as smooth SNMF (SSNMF), and examine how it relates to SNMF and orthogonal NMF. We then propose a convex model for SSNMF and show that it provably recovers the sought-after factors, even in the presence of noise. We finally adapt an existing fast gradient method to solve this convex model for SSNMF, and show that it compares favorably with state-of-the-art methods on both synthetic and hyperspectral datasets.

math.NA↗

arXiv2025

Robustness of Minimum-Volume Nonnegative Matrix Factorization under an Expanded Sufficiently Scattered Condition

Minimum-volume nonnegative matrix factorization (min-vol NMF) has been used successfully in many applications, such as hyperspectral imaging, chemical kinetics, spectroscopy, topic modeling, and audio source separation. However, its robustness to noise has been a long-standing open problem. In this paper, we prove that min-vol NMF identifies the groundtruth factors in the presence of noise under a condition referred to as the expanded sufficiently scattered condition which requires the data points to be sufficiently well scattered in the latent simplex generated by the basis vectors.

stat.ML↗

arXiv2025

Computing the nearest $\Omega$-admissible descriptor dissipative Hamiltonian system

For a given set $\Omega \subseteq \mathbb{C}$, a matrix pair $(E,A)$ is called $\Omega$-admissible if it is regular, impulse-free and its eigenvalues lie inside the region $\Omega$. In this paper, we provide a dissipative Hamiltonian characterization for the matrix pairs that are $\Omega$-admissible where $\Omega$ is an LMI region. We then use these results for solving the nearest $\Omega$-admissible matrix pair problem: Given a matrix pair $(E,A)$, find the nearest $\Omega$-admissible pair $(\tilde E, \tilde A)$ to the given pair $(E,A)$. We illustrate our results on several data sets and compare with the state of the art.

math.NA↗

arXiv2025

From latent factors to language: a user study on LLM-generated explanations for an inherently interpretable matrix-based recommender system

We investigate whether large language models (LLMs) can generate effective, user-facing explanations from a mathematically interpretable recommendation model. The model is based on constrained matrix factorization, where user types are explicitly represented and predicted item scores share the same scale as observed ratings, making the model's internal representations and predicted scores directly interpretable. This structure is translated into natural language explanations using carefully designed LLM prompts. Many works in explainable AI rely on automatic evaluation metrics, which often fail to capture users' actual needs and perceptions. In contrast, we adopt a user-centered approach: we conduct a study with 326 participants who assessed the quality of the explanations across five key dimensions-transparency, effectiveness, persuasion, trust, and satisfaction-as well as the recommendations themselves. To evaluate how different explanation strategies are perceived, we generate multiple explanation types from the same underlying model, varying the input information provided to the LLM. Our analysis reveals that all explanation types are generally well received, with moderate statistical differences between strategies. User comments further underscore how participants react to each type of explanation, offering complementary insights beyond the quantitative results.

cs.AI↗

arXiv2025

Identifiability of Nonnegative Tucker Decompositions -- Part I: Theory

Tensor decompositions have become a central tool in data science, with applications in areas such as data analysis, signal processing, and machine learning. A key property of many tensor decompositions, such as the canonical polyadic decomposition, is identifiability: the factors are unique, up to trivial scaling and permutation ambiguities. This allows one to recover the groundtruth sources that generated the data. The Tucker decomposition (TD) is a central and widely used tensor decomposition model. However, it is in general not identifiable. In this paper, we study the identifiability of the nonnegative TD (nTD). By adapting and extending identifiability results of nonnegative matrix factorization (NMF), we provide uniqueness results for nTD. Our results require the nonnegative matrix factors to have some degree of sparsity (namely, satisfy the separability condition, or the sufficiently scattered condition), while the core tensor only needs to have some slices (or linear combinations of them) or unfoldings with full column rank (but does not need to be nonnegative). Under such conditions, we derive several procedures, using either unfoldings or slices of the input tensor, to obtain identifiable nTDs by minimizing the volume of unfoldings or slices of the core tensor.

math.NA↗

arXiv2025

Efficient algorithms for the Hadamard decomposition

The Hadamard decomposition is a powerful technique for data analysis and matrix compression, which decomposes a given matrix into the element-wise product of two or more low-rank matrices. In this paper, we develop an efficient algorithm to solve this problem, leveraging an alternating optimization approach that decomposes the global non-convex problem into a series of convex sub-problems. To improve performance, we explore advanced initialization strategies inspired by the singular value decomposition (SVD) and incorporate acceleration techniques by introducing momentum-based updates. Beyond optimizing the two-matrix case, we also extend the Hadamard decomposition framework to support more than two low-rank matrices, enabling approximations with higher effective ranks while preserving computational efficiency. Finally, we conduct extensive experiments to compare our method with the existing gradient descent-based approaches for the Hadamard decomposition and with traditional low-rank approximation techniques. The results highlight the effectiveness of our proposed method across diverse datasets.

cs.LG↗

arXiv2025

Computing cone-constrained singular values of matrices

This paper deals with the numerical computation of the least singular value of a rectangular matrix $A$ relative to a pair of closed convex cones $(P,Q)$, which is defined as the optimal value of the non-convex optimization problem of minimizing $\langle u,Av\rangle$ such that $u$ and $v$ are unit vectors in $P$ and $Q$, respectively. When $A$ is the identity matrix, the least singular value coincides with the cosine of the largest angle between $P$ and $Q$. When $P$ and $Q$ are positive orthants, the least singular value is called the least Pareto singular value of $A$ and has applications, for instance, in graph theory. We prove the NP-hardness of all the above problems, while identifying cases when such problems can be solved in polynomial time. We then propose four algorithms. Two are exact algorithms, meaning that they are guaranteed to compute a globally optimal solution; one uses an exact non-convex quadratic programming solver, and the other a brute-force active-set method. The other two are heuristics, meaning that they rapidly compute locally optimal solutions; one uses an alternating projection algorithm with extrapolation, and the other a sequential partial linearization approach based on fractional programming. We illustrate the use of these algorithms on several examples.

math.OC↗

arXiv2025

An extrapolated and provably convergent algorithm for nonlinear matrix decomposition with the ReLU function

ReLU matrix decomposition (RMD) is the following problem: given a sparse, nonnegative matrix $X$ and a factorization rank $r$, identify a rank-$r$ matrix $\Theta$ such that $X\approx \max(0,\Theta)$. RMD is a particular instance of nonlinear matrix decomposition (NMD) that finds application in data compression, matrix completion with entries missing not at random, and manifold learning. The standard RMD model minimizes the least squares error, that is, $\|X - \max(0,\Theta)\|_F^2$. The corresponding optimization problem, Least-Squares RMD (LS-RMD), is nondifferentiable and highly nonconvex. This motivated Saul to propose an alternative model, \revise{dubbed Latent-RMD}, where a latent variable $Z$ is introduced and satisfies $\max(0,Z)=X$ while minimizing $\|Z - \Theta\|_F^2$ (``A nonlinear matrix decomposition for mining the zeros of sparse data'', SIAM J.\ Math.\ Data Sci., 2022). Our first contribution is to show that the two formulations may yield different low-rank solutions $\Theta$. We then consider a reparametrization of the Latent-RMD, called 3B-RMD, in which $\Theta$ is substituted by a low-rank product $WH$, where $W$ has $r$ columns and $H$ has $r$ rows. Our second contribution is to prove the convergence of a block coordinate descent (BCD) approach applied to 3B-RMD. Our third contribution is a novel extrapolated variant of BCD, dubbed eBCD, which we prove is also convergent under mild assumptions. We illustrate the significant acceleration effect of eBCD compared to eBCD, and also show that eBCD performs well against the state of the art on synthetic and real-world data sets.

cs.LG↗