arXiv subjects

Giovanni Seraghiti

Publications and source records attributed to Giovanni Seraghiti.

5 recordsLinked to original sources

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 analyze, under simplified probabilistic assumptions, how the sparsity in the data enforces zero solution in the optimal scalar update in the factors of L1-NMF when all the other entries are kept fixed. This provides an intuition of the connection between the sparsity of the L1-NMF factors with the sparsity of the input. Even though sparsity favors interpretability, 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. Although it lacks convergence guarantees to a stationary point, sCD is, to the best of our knowledge, 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, including imaging mass spectrometry and topic modeling, to show the effectiveness of our new proposed model (wL1-NMF) and algorithm (sCD).

cs.LG↗

arXiv2026

bAdag: an adaptive block coordinate gradient method for smooth nonconvex functions

A new Block Coordinate Gradient (BCG) method, dubbed bAdag, for smooth, nonconvex minimization problem is proposed; it falls in the class of Objective Function Free Optimization (OFFO) methods, and it is based on the AdaGrad algorithm. At each iteration, our method computes an adaptive step size based on the cumulative sum of block gradients, instead of full gradients as in AdaGrad-type methods. We prove ergodic, sublinear convergence rates for the bAdag algorithm when minimizing a smooth, possibly nonconvex objective under the (block) Lipschitz continuity assumption on the gradient. Our theory covers three widely popular block selection strategies: the Cyclic (C) rule, Uniform Random selection (UR), and the greedy Gauss-Southwell (GS) rule. We also extend our algorithm and its convergence theory to box-constrained smooth functions. We validate the proposed algorithms through synthetic and real-world experiments.

math.OC↗

arXiv2026

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 $Θ$ such that $X\approx \max(0,Θ)$. 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, $\|X - \max(0,Θ)\|_F^2$. 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 $\|Z - Θ\|_F^2$ (``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 $Θ$. We then consider a reparametrization of the Latent-RMD, called 3B-RMD, in which $Θ$ 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↗

arXiv2025

prunAdag: an adaptive pruning-aware gradient method

A pruning-aware adaptive gradient method is proposed which classifies the variables in two sets before updating them using different strategies. This technique extends the ``relevant/irrelevant" approach of Ding (2019) and Zimmer et al. (2022) and allows a posteriori sparsification of the solution of model parameter fitting problems. The new method is proved to be convergent with a global rate of decrease of the averaged gradient's norm of the form $\calO(\log(k)/\sqrt{k+1})$. Numerical experiments on several applications show that it is competitive.

math.OC↗

arXiv2023

Accelerated Algorithms for Nonlinear Matrix Decomposition with the ReLU function

In this paper, we study the following nonlinear matrix decomposition (NMD) problem: given a sparse nonnegative matrix $X$, find a low-rank matrix $Θ$ such that $X \approx f(Θ)$, where $f$ is an element-wise nonlinear function. We focus on the case where $f(\cdot) = \max(0, \cdot)$, the rectified unit (ReLU) non-linear activation. We refer to the corresponding problem as ReLU-NMD. We first provide a brief overview of the existing approaches that were developed to tackle ReLU-NMD. Then we introduce two new algorithms: (1) aggressive accelerated NMD (A-NMD) which uses an adaptive Nesterov extrapolation to accelerate an existing algorithm, and (2) three-block NMD (3B-NMD) which parametrizes $Θ= WH$ and leads to a significant reduction in the computational cost. We also propose an effective initialization strategy based on the nuclear norm as a proxy for the rank function. We illustrate the effectiveness of the proposed algorithms (available on gitlab) on synthetic and real-world data sets.

cs.LG↗