A Unified Convergence Analysis of the Multiplicative Update Algorithm for Regularized Nonnegative Matrix Factorization
The multiplicative update (MU) algorithm has been extensively used to estimate the basis and coefficient matrices in nonnegative matrix factorization (NMF) problems under a wide range of divergences and regularizers. However, theoretical convergence guarantees have only been derived for a few special divergences without regularization. In this work, we provide a conceptually simple, self-contained, and unified proof for the convergence of the MU algorithm applied on NMF with a wide range of divergences and regularizers. Our main result shows the sequence of iterates (i.e., pairs of basis and coefficient matrices) produced by the MU algorithm converges to the set of stationary points of the non-convex NMF optimization problem. Our proof strategy has the potential to open up new avenues for analyzing similar problems in machine learning and signal processing.
Code (0)
등록된 구현이 없습니다.
Similar Papers 제목 키워드 기반
Stochastic variance reduced multiplicative update for nonnegative matrix factorization
Nonnegative matrix factorization (NMF), a dimensionality reduction and factor analysis method, is a special case in which factor matrices have low-rank nonnegative constraints. Considering the stochastic learning in NMF,…
Dimensionality ReductionLangevin Multiplicative Weights Update with Applications in Polynomial Portfolio Management
We consider nonconvex optimization problem over simplex, and more generally, a product of simplices. We provide an algorithm, Langevin Multiplicative Weights Update (LMWU) for solving global optimization problems by addi…
global-optimizationManagementGraph Matching via Multiplicative Update Algorithm
As a fundamental problem in computer vision, graph matching problem can usually be formulated as a Quadratic Programming (QP) problem with doubly stochastic and discrete (integer) constraints. Since it is NP-hard, approx…
Graph MatchingA Non-Negative Matrix Factorization Game
We present a novel game-theoretic formulation of Non-Negative Matrix Factorization (NNMF), a popular data-analysis method with many scientific and engineering applications. The game-theoretic formulation is shown to have…
Last-Iterate Convergence with Full and Noisy Feedback in Two-Player Zero-Sum Games
This paper proposes Mutation-Driven Multiplicative Weights Update (M2WU) for learning an equilibrium in two-player zero-sum normal-form games and proves that it exhibits the last-iterate convergence property in both full…
Multi-agent Reinforcement Learning