Symmetric nonnegative matrix factorization (SNMF) is equivalent to computing a symmetric nonnegative low rank approximation of a data similarity matrix. It inherits the good data interpretability of the well-known nonnegative matrix factorization technique and have better ability of clustering nonlinearly separable data. In this paper, we focus on the algorithmic aspect of the SNMF problem and propose simple inexact block coordinate decent methods to address the problem, leading to both serial and parallel algorithms. The proposed algorithms have guaranteed stationary convergence and can efficiently handle large-scale and/or sparse SNMF problems. Extensive simulations verify the effectiveness of the proposed algorithms compared to recent state-of-the-art algorithms.
This paper is concerned with the inverse problem of constructing a symmetric nonnegative matrix from realizable spectrum. We reformulate the inverse problem as an underdetermined nonlinear matrix equation over a Riemannian product manifold. To solve it, we develop a Riemannian underdetermined inexact Newton dogleg method for solving a general underdetermined nonlinear equation defined between Riemannian manifolds and Euclidean spaces. The global and quadratic convergence of the proposed method is established under some mild assumptions. Then we solve the inverse problem by applying the propose…
The self regulation problem as an inexact steepest descent method for multicriteria optimization
IME-Universidade Federal de Goiás（戈亚斯联邦大学数学与工程学院）；DM, Universidade Federal do Piauí（皮奥伊联邦大学数学系）；COPPE/Sistemas-Universidade Federal do Rio de Janeiro（里约热内卢联邦大学科佩系统工程研究所）；Aix-Marseille University（艾克斯-马赛大学）
In this paper we introduce a novel abstract descent scheme suited for the minimization of proper and lower semicontinuous functions. The proposed abstract scheme generalizes a set of properties that are crucial for the convergence of several first-order methods designed for nonsmooth nonconvex optimization problems. Such properties guarantee the convergence of the full sequence of iterates to a stationary point, if the objective function satisfies the Kurdyka-Lojasiewicz property. The abstract framework allows for the design of new algorithms. We propose two inertial-type algorithms with imple…
University of Edinburgh（爱丁堡大学）；King Abdullah University of Science and Technology (KAUST)（沙特国王 Abdullah 科技大学 (KAUST)）；MIPT（莫斯科物理技术学院）
In this paper we present a convergence rate analysis of inexact variants of several randomized iterative methods. Among the methods studied are: stochastic gradient descent, stochastic Newton, stochastic proximal point and stochastic subspace ascent. A common feature of these methods is that in their update rule a certain sub-problem needs to be solved exactly. We relax this requirement by allowing for the sub-problem to be solved inexactly. In particular, we propose and analyze inexact randomized iterative methods for solving three closely related problems: a convex stochastic quadratic optim…
Université de Mons（蒙斯大学）；Université de 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（雷恩第一大学计算机科学研究所）
This paper is concerned with improving the empirical convergence speed of block-coordinate descent algorithms for approximate nonnegative tensor factorization (NTF). We propose an extrapolation strategy in-between block updates, referred to as heuristic extrapolation with restarts (HER). HER significantly accelerates the empirical convergence speed of most existing block-coordinate algorithms for dense NTF, in particular for challenging computational scenarios, while requiring a negligible additional computational budget.
We study unconstrained optimization problems with nonsmooth and convex objective function in the form of a mathematical expectation. The proposed method approximates the expected objective function with a sample average function using Inexact Restoration-based adapted sample sizes. The sample size is chosen in an adaptive manner based on Inexact Restoration. The algorithm uses line search and assumes descent directions with respect to the current approximate function. We prove the a.s. convergence under standard assumptions. Numerical results for two types of problems, machine learning loss fu…
In this paper, we consider smooth convex optimization problems with simple constraints and inexactness in the oracle information such as value, partial or directional derivatives of the objective function. We introduce a unifying framework, which allows to construct different types of accelerated randomized methods for such problems and to prove convergence rate theorems for them. We focus on accelerated random block-coordinate descent, accelerated random directional search, accelerated random derivative-free method and, using our framework, provide their versions for problems with inexact ora…
The full approximation storage (FAS) scheme is a widely used multigrid method for nonlinear problems. In this paper, a new framework to design and analyze FAS-like schemes for convex optimization problems is developed. The new method, the Fast Subspace Descent (FASD) scheme, which generalizes classical FAS, can be recast as an inexact version of nonlinear multigrid methods based on space decomposition and subspace correction. The local problem in each subspace can be simplified to be linear and one gradient descent iteration (with an appropriate step size) is enough to ensure a global linear (…
A convergence framework for inexact nonconvex and nonsmooth algorithms and its applications to several iterations
National University of Defense Technology（国防科技大学）；The State Key Laboratory for High Performance Computation, National University of Defense Technology（高性能计算国家重点实验室（国防科技大学））；School of Mathematics and Computational Science, Xiangtan University（湘潭大学数学与计算科学学院）
Symmetric nonnegative matrix factorization (symNMF) is a variant of nonnegative matrix factorization (NMF) that allows to handle symmetric input matrices and has been shown to be particularly well suited for clustering tasks. In this paper, we present a new model, dubbed off-diagonal symNMF (ODsymNMF), that does not take into account the diagonal entries of the input matrix in the objective function. ODsymNMF has three key advantages compared to symNMF. First, ODsymNMF is theoretically much more sound as there always exists an exact factorization of size at most $\nicefrac{n(n-1)}{2}$ where $n…
Backtracking is an inexact line search procedure that selects the first value in a sequence $x_0, x_0β, x_0β^2...$ that satisfies $g(x)\leq 0$ on $\mathbb{R}_+$ with $g(x)\leq 0$ iff $x\leq x^*$. This procedure is widely used in descent direction optimization algorithms with Armijo-type conditions. It both returns an estimate in $(βx^*,x^*]$ and enjoys an upper-bound $\lceil \log_β ε/x_0 \rceil$ on the number of function evaluations to terminate, with $ε$ a lower bound on $x^*$. The basic bracketing mechanism employed in several root-searching methods is adapted here for the purpose of perform…
Accelerated Parallel and Distributed Algorithm using Limited Internal Memory for Nonnegative Matrix Factorization
Japan Advanced Institute of Science and Technology（日本科学技术高等学院）；University of Engineering and Technology, Vietnam National University, Hanoi（越南国立大学河内大学工程技术学院）；John von Neumann Institute, Vietnam National University, Ho Chi Minh City（越南国立大学胡志明市冯·诺依曼研究所）
