An Efficient Newton Algorithm for Nonnegative Matrix Factorization with the Kullback-Leibler Divergence

• URL: http://arxiv.org/abs/2607.13919v1
• Date: Wed, 15 Jul 2026 14:59:07 GMT
• Title: An Efficient Newton Algorithm for Nonnegative Matrix Factorization with the Kullback-Leibler Divergence
• Abstract summary: Nonnegative Matrix Factorization (NMF) approximates a nonnegative matrix by the product of two low-rank nonnegative factors.<n>Most KL-NMF algorithms minimize a separable majorant of the loss to find their next decomposed.<n>We argue that this method has reached its limits and propose to use instead the second-order Taylor expansion of the loss.<n>This yields an efficient KL-NMF algorithm which provably converges and which competes favorably with state-of-the-art algorithms on a large variety of datasets.
• Score: 1.1470070927586018
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Nonnegative Matrix Factorization (NMF) is a fundamental tool in unsupervised learning, which approximates a nonnegative matrix by the product of two low-rank nonnegative factors. The Kullback-Leibler (KL) divergence is best suited to measure the data to model discrepancy when the decomposed data sample follows a Poisson distribution, which is the case for count datasets such as term-document matrices or images. Most KL-NMF algorithms in the literature minimize a separable majorant of the loss to find their next iterate. We argue that this method has reached its limits and propose to use instead the second-order Taylor expansion of the loss, leading to a Newton-type method. We minimize this non-separable surrogate by proposing a generalization of the well-known HALS algorithm. This yields an efficient KL-NMF algorithm which provably converges and which competes favorably with state-of-the-art algorithms on a large variety of datasets.

Related papers

• Spectral Entry-wise Matrix Estimation for Low-Rank Reinforcement Learning [53.445068584013896]
  We study matrix estimation problems arising in reinforcement learning (RL) with low-rank structure. In low-rank bandits, the matrix to be recovered specifies the expected arm rewards, and for low-rank Markov Decision Processes (MDPs), it may for example characterize the transition kernel of the MDP. We show that simple spectral-based matrix estimation approaches efficiently recover the singular subspaces of the matrix and exhibit nearly-minimal entry-wise error.
  arXiv  Detail & Related papers  (2023-10-10T17:06:41Z)
• A Second-Order Majorant Algorithm for Nonnegative Matrix Factorization [2.646309221150203]
  We introduce a general second-order optimization framework for NMF under both quadratic and $beta$-divergence loss functions.<n>Second-Order Majorant (SOM) constructs a local quadratic majorization of the loss function by majorizing its Hessian matrix.<n>We show that mSOM consistently outperforms state-of-the-art algorithms across multiple loss functions.
  arXiv  Detail & Related papers  (2023-03-31T12:09:36Z)
• Contaminated Images Recovery by Implementing Non-negative Matrix Factorisation [0.0]
  We theoretically examine the robustness of the traditional NMF, HCNMF, and L2,1-NMF algorithms and execute sets of experiments to demonstrate the robustness on ORL and Extended YaleB datasets. Due to the computational cost of these approaches, our final models, such as the HCNMF and L2,1-NMF model, fail to converge within the parameters of this work.
  arXiv  Detail & Related papers  (2022-11-08T13:50:27Z)
• Orthogonal Nonnegative Matrix Factorization with Sparsity Constraints [0.0]
  This article presents a novel approach to solving the sparsity-constrained Orthogonal Nonnegative Matrix Factorization (SCONMF) problem.<n>By reformulating SCONMF as a capacity-constrained facility-location problem, the proposed method naturally integrates non-negativity, orthogonality, and sparsity constraints.<n>Specifically, our approach integrates control-barrier function (CBF) based framework used for dynamic optimal control design problems with maximum-entropy-principle-based framework used for facility location problems to enforce these constraints while ensuring robust factorization.
  arXiv  Detail & Related papers  (2022-10-06T04:30:59Z)
• Log-based Sparse Nonnegative Matrix Factorization for Data Representation [55.72494900138061]
  Nonnegative matrix factorization (NMF) has been widely studied in recent years due to its effectiveness in representing nonnegative data with parts-based representations. We propose a new NMF method with log-norm imposed on the factor matrices to enhance the sparseness. A novel column-wisely sparse norm, named $ell_2,log$-(pseudo) norm, is proposed to enhance the robustness of the proposed method.
  arXiv  Detail & Related papers  (2022-04-22T11:38:10Z)
• Solving weakly supervised regression problem using low-rank manifold regularization [77.34726150561087]
  We solve a weakly supervised regression problem. Under "weakly" we understand that for some training points the labels are known, for some unknown, and for others uncertain due to the presence of random noise or other reasons such as lack of resources. In the numerical section, we applied the suggested method to artificial and real datasets using Monte-Carlo modeling.
  arXiv  Detail & Related papers  (2021-04-13T23:21:01Z)
• Self-supervised Symmetric Nonnegative Matrix Factorization [82.59905231819685]
  Symmetric nonnegative factor matrix (SNMF) has demonstrated to be a powerful method for data clustering. Inspired by ensemble clustering that aims to seek better clustering results, we propose self-supervised SNMF (S$3$NMF) We take advantage of the sensitivity to code characteristic of SNMF, without relying on any additional information.
  arXiv  Detail & Related papers  (2021-03-02T12:47:40Z)
• Nonnegative Matrix Factorization with Toeplitz Penalty [0.0]
  Nonnegative Matrix Factorization (NMF) is an unsupervised learning algorithm that produces a linear, parts-based approximation of a data matrix. We propose a new NMF algorithm that makes use of non-data dependent auxiliary constraints.
  arXiv  Detail & Related papers  (2020-12-07T13:49:23Z)
• Algorithms for Nonnegative Matrix Factorization with the Kullback-Leibler Divergence [20.671178429005973]
  Kullback-Leibler (KL) divergence is one of the most widely used objective function for nonnegative matrix factorization (NMF) We propose three new algorithms that guarantee the non-increasingness of the objective function. We conduct extensive numerical experiments to provide a comprehensive picture of the performances of the KL NMF algorithms.
  arXiv  Detail & Related papers  (2020-10-05T11:51:39Z)
• Robust Low-rank Matrix Completion via an Alternating Manifold Proximal Gradient Continuation Method [47.80060761046752]
  Robust low-rank matrix completion (RMC) has been studied extensively for computer vision, signal processing and machine learning applications. This problem aims to decompose a partially observed matrix into the superposition of a low-rank matrix and a sparse matrix, where the sparse matrix captures the grossly corrupted entries of the matrix. A widely used approach to tackle RMC is to consider a convex formulation, which minimizes the nuclear norm of the low-rank matrix (to promote low-rankness) and the l1 norm of the sparse matrix (to promote sparsity). In this paper, motivated by some recent works on low-
  arXiv  Detail & Related papers  (2020-08-18T04:46:22Z)
• Understanding Implicit Regularization in Over-Parameterized Single Index Model [55.41685740015095]
  We design regularization-free algorithms for the high-dimensional single index model. We provide theoretical guarantees for the induced implicit regularization phenomenon.
  arXiv  Detail & Related papers  (2020-07-16T13:27:47Z)
• A Block Coordinate Descent-based Projected Gradient Algorithm for Orthogonal Non-negative Matrix Factorization [0.0]
  This article utilizes the projected gradient method (PG) for a non-negative matrix factorization problem (NMF) We penalise the orthonormality constraints and apply the PG method via a block coordinate descent approach.
  arXiv  Detail & Related papers  (2020-03-23T13:24:43Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.