Nonconvex Optimization Tools for Large-Scale Matrix and Tensor Decomposition with Structured Factors
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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 algorithmic procedures for a small set of problems, e.g., nonnegativity or sparsity-constrained factorization, we take a 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 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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