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

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Document pages: 15 pages

Abstract: The proposed article aims at offering a comprehensive tutorial for thecomputational aspects of structured matrix and tensor factorization. Unlikeexisting tutorials that mainly focus on { it algorithmic procedures} for asmall set of problems, e.g., nonnegativity or sparsity-constrainedfactorization, we take a { it top-down} approach: we start with generaloptimization theory (e.g., inexact and accelerated block coordinate descent,stochastic optimization, and Gauss-Newton methods) that covers a wide range offactorization problems with diverse constraints and regularization terms ofengineering interest. Then, we go `under the hood to showcase specificalgorithm design under these introduced principles. We pay a particularattention to recent algorithmic developments in structured tensor and matrixfactorization (e.g., random sketching and adaptive step size based stochasticoptimization and structure-exploiting second-order algorithms), which are thestate of the art---yet much less touched upon in the literature compared to{ it block coordinate descent} (BCD)-based methods. We expect that the articleto have an educational values in the field of structured factorization and hopeto stimulate more research in this important and exciting direction.

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