An alternating direction and projection algorithm for structure-enforced matrix factorization

• Published in: Computational optimization and applications Vol. 68; no. 2; pp. 333 - 362
• Main Authors: Xu, Lijun, Yu, Bo, Zhang, Yin
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.11.2017; Springer Nature B.V
• Subjects: Algorithms; Combinatorial analysis; Convex and Discrete Geometry; Dictionaries; Factorization; Machine learning; Management Science; Mathematical models; Mathematics; Mathematics and Statistics; Matrix methods; Operations Research; Operations Research/Decision Theory; Optimization; Principal components analysis; Signal processing; Statistics; Alternating direction method; Sparse optimization; Projection; Matrix factorization; Adaptive penalty parameter; Dictionary learning
• ISSN: 0926-6003; 1573-2894
• DOI: 10.1007/s10589-017-9913-x

Structure-enforced matrix factorization (SeMF) represents a large class of mathematical models appearing in various forms of principal component analysis, sparse coding, dictionary learning and other machine learning techniques useful in many applications including neuroscience and signal processing. In this paper, we present a unified algorithm framework, based on the classic alternating direction method of multipliers (ADMM), for solving a wide range of SeMF problems whose constraint sets permit low-complexity projections. We propose a strategy to adaptively adjust the penalty parameters which is the key to achieving good performance for ADMM. We conduct extensive numerical experiments to compare the proposed algorithm with a number of state-of-the-art special-purpose algorithms on test problems including dictionary learning for sparse representation and sparse nonnegative matrix factorization. Results show that our unified SeMF algorithm can solve different types of factorization problems as reliably and as efficiently as special-purpose algorithms. In particular, our SeMF algorithm provides the ability to explicitly enforce various combinatorial sparsity patterns that, to our knowledge, has not been considered in existing approaches.