Computing Large-Scale Matrix and Tensor Decomposition with Structured Factors: A Unified Nonconvex Optimization Perspective

• URL: http://arxiv.org/abs/2006.08183v2
• Date: Wed, 5 Aug 2020 20:31:41 GMT
• Title: Computing Large-Scale Matrix and Tensor Decomposition with Structured Factors: A Unified Nonconvex Optimization Perspective
• Authors: Xiao Fu, Nico Vervliet, Lieven De Lathauwer, Kejun Huang, Nicolas Gillis
• Abstract summary: This article aims at offering a comprehensive tutorial for the computational aspects of structured matrix and tensor factorization. We start with general optimization theory that covers a wide range of factorization problems with diverse constraints. Then, we go under the hood' to showcase specific algorithm design under these introduced principles.
• Score: 33.19643734230432
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The proposed article aims at offering a comprehensive tutorial for the computational aspects of structured matrix and tensor factorization. Unlike existing tutorials that mainly focus on {\it algorithmic procedures} for a small set of problems, e.g., nonnegativity or sparsity-constrained factorization, we take a {\it top-down} approach: we start with general optimization theory (e.g., inexact and accelerated block coordinate descent, stochastic optimization, and Gauss-Newton methods) that covers a wide range of factorization problems with diverse constraints and regularization terms of engineering interest. Then, we go `under the hood' to showcase specific algorithm design under these introduced principles. We pay a particular attention to recent algorithmic developments in structured tensor and matrix factorization (e.g., random sketching and adaptive step size based stochastic optimization and structure-exploiting second-order algorithms), which are the state of the art---yet much less touched upon in the literature compared to {\it block coordinate descent} (BCD)-based methods. We expect that the article to have an educational values in the field of structured factorization and hope to stimulate more research in this important and exciting direction.

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