In this paper, we study sparse factorization of the (scaled) square all-ones matrix $J$ of arbitrary order. We introduce the concept of hierarchically banded matrices and propose two types of hierarchically banded factorization of $J$: the reduced hierarchically banded (RHB) factorization and the doubly stochastic hierarchically banded (DSHB) factorization. Based on the DSHB factorization, we propose the sequential doubly stochastic (SDS) factorization, in which~$J$ is decomposed as a product of sparse, doubly stochastic matrices. Finally, we discuss the application of the proposed sparse factorizations to the decentralized average consensus problem and decentralized optimization.
Using Underapproximations for Sparse Nonnegative Matrix Factorization
非负矩阵分解（Nonnegative Matrix Factorization）是将一个非负数据矩阵（nonnegative data matrix）近似分解为两个低秩非负矩阵的乘积。它已成功地应用于许多领域的数据分析技术，例如，文本挖掘，图像处理，微阵列数据分析，协同过滤等，我们介绍了一种新的方法来解决NMF问题，基于使用的欠近似技术，并显示其有效性，以获得稀疏的解决方案。这种方法，基于拉格朗日松弛，允许NMF问题的递归方式的决议。我们还证明了下近似问题是NP-困难的任何固定的分解秩，使用减少的最大边biclique问题的二部图。我们在几个标准图像数据集上测试了我们的欠近似方法的两个变体，并表明它们提供了具有低重建误差的稀疏部分表示。我们的结果是可比的，有时优于两个标准的稀疏非负矩阵分解技术。
An Alternating Minimization Algorithm for Federated Sparse Orthogonal Nonnegative Matrix Factorization
School of Intelligent Systems Engineering, Sun Yat-sen University（中山大学智能系统工程学院）；School of Mechatronic Engineering and Automation, Shanghai University（上海大学机电工程与自动化学院）
Mixed membership factorization is a popular approach for analyzing data sets that have within-sample heterogeneity. In recent years, several algorithms have been developed for mixed membership matrix factorization, but they only guarantee estimates from a local optimum. Here, we derive a global optimization (GOP) algorithm that provides a guaranteed $ε$-global optimum for a sparse mixed membership matrix factorization problem. We test the algorithm on simulated data and find the algorithm always bounds the global optimum across random initializations and explores multiple modes efficiently.
Online Learning for Matrix Factorization and Sparse Coding
INRIA（法国国家信息与自动化研究所）；Laboratoire d’Informatique de l’Ecole Normale Supérieure（巴黎高等师范学院信息学实验室）；Ecole Normale Supérieure（巴黎高等师范学院）；University of Minnesota（明尼苏达大学）
University of Mons（蒙斯大学）；Univ Rennes（雷恩大学）；Inria（法国国家信息与自动化研究所）；CNRS（法国国家科学研究中心）；IRISA（法国雷恩信息与自动化系统研究所）
We propose a new variant of nonnegative matrix factorization (NMF), combining separability and sparsity assumptions. Separability requires that the columns of the first NMF factor are equal to columns of the input matrix, while sparsity requires that the columns of the second NMF factor are sparse. We call this variant sparse separable NMF (SSNMF), which we prove to be NP-complete, as opposed to separable NMF which can be solved in polynomial time. The main motivation to consider this new model is to handle underdetermined blind source separation problems, such as multispectral image unmixing.…
Hyperspectral Anomaly Detection via Group Sparse Low-Rank Tensor Factorization With Automatic Anomaly Grouping
Central China Normal University（华中师范大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）
Sparse Hyperparametric Itakura-Saito Nonnegative Matrix Factorization via Bi-Level Optimization
organization= Water Research Institute - National Research Council (IRSA-CNR) , country= Italy；organization= Department of Mathematics, University of Bari Aldo Moro , country= Italy；organization= School of Electronics；Computer Science, University of Southampton , country= UK；organization= Faculty of Electronics, Photonics；Microsystems, Wroc aw University of Science
Matrix decomposition is ubiquitous and has applications in various fields like speech processing, data mining and image processing to name a few. Under matrix decomposition, nonnegative matrix factorization is used to decompose a nonnegative matrix into a product of two nonnegative matrices which gives some meaningful interpretation of the data. Thus, nonnegative matrix factorization has an edge over the other decomposition techniques. In this paper, we propose two novel iterative algorithms based on Majorization Minimization (MM)-in which we formulate a novel upper bound and minimize it to ge…
Modern-day seismic imaging and monitoring technology increasingly rely on dense full-azimuth sampling. Unfortunately, the costs of acquiring densely sampled data rapidly become prohibitive and we need to look for ways to sparsely collect data, e.g. from sparsely distributed ocean bottom nodes, from which we then derive densely sampled surveys through the method of wavefield reconstruction. Because of their relatively cheap and simple calculations, wavefield reconstruction via matrix factorizations has proven to be a viable and scalable alternative to the more generally used transform-based met…
We propose to compute a sparse approximate inverse Cholesky factor $L$ of a dense covariance matrix $Θ$ by minimizing the Kullback-Leibler divergence between the Gaussian distributions $\mathcal{N}(0, Θ)$ and $\mathcal{N}(0, L^{-\top} L^{-1})$, subject to a sparsity constraint. Surprisingly, this problem has a closed-form solution that can be computed efficiently, recovering the popular Vecchia approximation in spatial statistics. Based on recent results on the approximate sparsity of inverse Cholesky factors of $Θ$ obtained from pairwise evaluation of Green's functions of elliptic boundary-va…
