Tensor factorization has proven useful in a wide range of applications, from sensor array processing to communications, speech and audio signal processing, and machine learning. With few recent exceptions, all tensor factorization algorithms were originally developed for centralized, in-memory computation on a single machine; and the few that break away from this mold do not easily incorporate practically important constraints, such as nonnegativity. A new constrained tensor factorization framework is proposed in this paper, building upon the Alternating Direction method of Multipliers (ADMoM). It is shown that this simplifies computations, bypassing the need to solve constrained optimization problems in each iteration; and it naturally leads to distributed algorithms suitable for parallel implementation on regular high-performance computing (e.g., mesh) architectures. This opens the door for many emerging big data-enabled applications. The methodology is exemplified using nonnegativity as a baseline constraint, but the proposed framework can more-or-less readily incorporate many other types of constraints. Numerical experiments are very encouraging, indicating that the ADMoM-based nonnegative tensor factorization (NTF) has high potential as an alternative to state-of-the-art approaches.
In this paper, we consider a prototypical convex optimization problem with multi-block variables and separable structures. By adding the Logarithmic Quadratic Proximal (LQP) regularizer with suitable proximal parameter to each of the first grouped subproblems, we develop a partial LQP-based Alternating Direction Method of Multipliers (ADMM-LQP). The dual variable is updated twice with relatively larger stepsizes than the classical region $(0,\frac{1+\sqrt{5}}{2})$. Using a prediction-correction approach to analyze properties of the iterates generated by ADMM-LQP, we establish its global conver…
A method for quasistatic cohesive fracture is introduced that uses an alternating direction method of multipliers (ADMM) to implement an energy approach to cohesive fracture. The ADMM algorithm minimizes a non-smooth, non-convex potential functional at each strain increment to predict the evolution of a cohesive-elastic system. The optimization problem bypasses the explicit stress criterion of force-based (Newtonian) methods, which interferes with Newton iterations impeding convergence. The model is extended with an extrapolation method that significantly reduces the computation time of the se…
Iteratively Linearized Reweighted Alternating Direction Method of Multipliers for a Class of Nonconvex Problems
Department of Mathematics, National University of Defense Technology（国防科技大学数学系）；College of Computer, National University of Defense Technology（国防科技大学计算机学院）；The State Key Laboratory for High Performance Computation, National University of Defense Technology（国防科技大学高性能计算国家重点实验室）；Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University（湖南计算与模拟科学工程重点实验室，湘潭大学数学与计算科学学院）
We propose two numerical algorithms in the fully nonconvex setting for the minimization of the sum of a smooth function and the composition of a nonsmooth function with a linear operator. The iterative schemes are formulated in the spirit of the proximal alternating direction method of multipliers and its linearized variant, respectively. The proximal terms are introduced via variable metrics, a fact which allows us to derive new proximal splitting algorithms for nonconvex structured optimization problems, as particular instances of the general schemes. Under mild conditions on the sequence of…
This paper introduces an algorithm for the nonnegative matrix factorization-and-completion problem, which aims to find nonnegative low-rank matrices X and Y so that the product XY approximates a nonnegative data matrix M whose elements are partially known (to a certain accuracy). This problem aggregates two existing problems: (i) nonnegative matrix factorization where all entries of M are given, and (ii) low-rank matrix completion where nonnegativity is not required. By taking the advantages of both nonnegativity and low-rankness, one can generally obtain superior results than those of just us…
We study several versions of the alternating direction method of multipliers (ADMM) for solving the convex problem of finding the distance between two ellipsoids and the nonconvex problem of finding the distance between the boundaries of two ellipsoids. In the convex case we present the ADMM with and without automatic penalty updates and demonstrate via numerical experiments on problems of various dimensions that our methods significantly outperform all other existing methods for finding the distance between ellipsoids. In the nonconvex case we propose a heuristic rule for updating the penalty…
We introduce and analyze BPALM and A-BPALM, two multi-block proximal alternating linearized minimization algorithms using Bregman distances for solving structured nonconvex problems. The objective function is the sum of a multi-block relatively smooth function (i.e., relatively smooth by fixing all the blocks except one) and block separable (nonsmooth) nonconvex functions. It turns out that the sequences generated by our algorithms are subsequentially convergent to critical points of the objective function, while they are globally convergent under KL inequality assumption. Further, the rate of…
Nonnegative matrix factorization (NMF) is a prominent technique for data dimensionality reduction that has been widely used for text mining, computer vision, pattern discovery, and bioinformatics. In this paper, a framework called ARkNLS (Alternating Rank-k Nonnegativity constrained Least Squares) is proposed for computing NMF. First, a recursive formula for the solution of the rank-k nonnegativity-constrained least squares (NLS) is established. This recursive formula can be used to derive the closed-form solution for the Rank-k NLS problem for any integer k $\ge$ 1. As a result, each subprobl…
A Dual Symmetric Gauss-Seidel Alternating Direction Method of Multipliers for Hyperspectral Sparse Unmixing
School of Information Science and technology, and the Provincial Key Lab of Information Coding and Trans- mission, Southwest Jiaotong University（信息科学与技术学院，信息编码与传输省重点实验室，西南交通大学）；School of Mathematics, Southwest Jiaotong University（数学学院，西南交通大学）；School of Computer and Computing Science, Zhejiang University City College（计算机与计算科学学院，浙江大学城市学院）
