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Positive Semidefinite Matrix Factorization A Connection with Phase Retrieval and Affine Rank Minimization

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Document pages: 18 pages

Abstract: Positive semidefinite matrix factorization (PSDMF) expresses each entry of anonnegative matrix as the inner product of two positive semidefinite (psd)matrices. When all these psd matrices are constrained to be diagonal, thismodel is equivalent to nonnegative matrix factorization. Applications includecombinatorial optimization, quantum-based statistical models, and recommendersystems, among others. However, despite the increasing interest in PSDMF, onlya few PSDMF algorithms were proposed in the literature. In this work, weprovide a collection of tools for PSDMF, by showing that PSDMF algorithms canbe designed based on phase retrieval (PR) and affine rank minimization (ARM)algorithms. This procedure allows a shortcut in designing new PSDMF algorithms,as it allows to leverage some of the useful numerical properties of existing PRand ARM methods to the PSDMF framework. Motivated by this idea, we introduce anew family of PSDMF algorithms based on iterative hard thresholding (IHT). Thisfamily subsumes previously-proposed projected gradient PSDMF methods. We showthat there is high variability among PSDMF optimization problems that makes itbeneficial to try a number of methods based on different principles to tackledifficult problems. In certain cases, our proposed methods are the onlyalgorithms able to find a solution. In certain other cases, they convergefaster. Our results support our claim that the PSDMF framework can inheritdesired numerical properties from PR and ARM algorithms, leading to moreefficient PSDMF algorithms, and motivate further study of the links betweenthese models.

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