A Framework of Inertial Alternating Direction Method of Multipliers for Non-Convex Non-Smooth Optimization

In this paper, we propose an algorithmic framework dubbed inertial alternating direction methods of multipliers (iADMM), for solving a class of nonconvex nonsmooth multiblock composite optimization problems with linear constraints. Our framework employs the general minimization-majorization (MM) principle to update each block of variables so as to not only unify the convergence analysis of previous … Read more

An Inertial Block Majorization Minimization Framework for Nonsmooth Nonconvex Optimization

In this paper, we introduce TITAN, a novel inerTial block majorIzation minimization framework for non-smooth non-convex opTimizAtioN problems. TITAN is a block coordinate method (BCM) that embeds inertial force to each majorization-minimization step of the block updates. The inertial force is obtained via an extrapolation operator that subsumes heavy-ball and Nesterov-type accelerations for block proximal … Read more

Inertial Block Mirror Descent Method for Non-Convex Non-Smooth Optimization

In this paper, we propose inertial versions of block coordinate descent methods for solving non-convex non-smooth composite optimization problems. We use the general framework of Bregman distance functions to compute the proximal maps. Our method not only allows using two different extrapolation points to evaluate gradients and adding the inertial force, but also takes advantage … Read more

A semi-analytical approach for the positive semidefinite Procrustes problem

The positive semidefinite Procrustes (PSDP) problem is the following: given rectangular matrices $X$ and $B$, find the symmetric positive semidefinite matrix $A$ that minimizes the Frobenius norm of $AX-B$. No general procedure is known that gives an exact solution. In this paper, we present a semi-analytical approach to solve the PSDP problem. First, we characterize … Read more

Semidefinite Programming Based Preconditioning for More Robust Near-Separable Nonnegative Matrix Factorization

Nonnegative matrix factorization (NMF) under the separability assumption can provably be solved efficiently, even in the presence of noise, and has been shown to be a powerful technique in document classification and hyperspectral unmixing. This problem is referred to as near-separable NMF and requires that there exists a cone spanned by a small subset of … Read more

Robust Near-Separable Nonnegative Matrix Factorization Using Linear Optimization

Nonnegative matrix factorization (NMF) has been shown recently to be tractable under the separability assumption, under which all the columns of the input data matrix belong to the convex cone generated by only a few of these columns. Bittorf, Recht, R\’e and Tropp (`Factoring nonnegative matrices with linear programs’, NIPS 2012) proposed a linear programming … Read more

A Continuous Characterization of the Maximum-Edge Biclique Problem

The problem of finding large complete subgraphs in bipartite graphs (that is, bicliques) is a well-known combinatorial optimization problem referred to as the maximum-edge biclique problem (MBP), and has many applications, e.g., in web community discovery, biological data analysis and text mining. In this paper, we present a new continuous characterization for MBP. Given a … Read more

Robustness Analysis of HottTopixx, a Linear Programming Model for Factoring Nonnegative Matrices

Although nonnegative matrix factorization (NMF) is NP-hard in general, it has been shown very recently that it is tractable under the assumption that the input nonnegative data matrix is separable (that is, there exists a cone spanned by a small subset of the columns containing all columns). Since then, several algorithms have been designed to … Read more

Fast and Robust Recursive Algorithms for Separable Nonnegative Matrix Factorization

In this paper, we study the nonnegative matrix factorization problem under the separability assumption (that is, there exists a cone spanned by a small subset of the columns of the input nonnegative data matrix containing all columns), which is equivalent to the hyperspectral unmixing problem under the linear mixing model and the pure-pixel assumption. We … Read more