2018
DOI: 10.1007/s10589-018-9998-x
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Algorithms for positive semidefinite factorization

Abstract: This paper considers the problem of positive semidefinite factorization (PSD factorization), a generalization of exact nonnegative matrix factorization. Given an m-by-n nonnegative matrix X and an integer k, the PSD factorization problem consists in finding, if possible, symmetric k-by-k positive semidefinite matrices {A 1 , ..., A m } and {B 1 , ..., B n } such that Xi,j = trace(A i B j ) for i = 1, ..., m, and j = 1, ..., n. PSD factorization is NP-hard. In this work, we introduce several local optimization … Show more

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Cited by 12 publications
(60 citation statements)
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“…Our proposed SVPbased PSDMF algorithms subsume the projected gradient method (PGM) [3] by allowing the use of inner ranks smaller than K. We also show that SVP subsumes PRIME-Power [26], which is a majorization-minimization (MM)-based PR method. We further propose two variants to our basic SVP-based PSDMF-the first is fast singular value projection (FSVP), which is based on Nesterov's accelerated gradient descent [31] and subsumes the fast projected gradient method (FPGM) [3]. FPGM and FSVP require an additional parameter the user has to fine-tune to achieve sufficient acceleration.…”
Section: B Main Contributionsmentioning
confidence: 90%
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“…Our proposed SVPbased PSDMF algorithms subsume the projected gradient method (PGM) [3] by allowing the use of inner ranks smaller than K. We also show that SVP subsumes PRIME-Power [26], which is a majorization-minimization (MM)-based PR method. We further propose two variants to our basic SVP-based PSDMF-the first is fast singular value projection (FSVP), which is based on Nesterov's accelerated gradient descent [31] and subsumes the fast projected gradient method (FPGM) [3]. FPGM and FSVP require an additional parameter the user has to fine-tune to achieve sufficient acceleration.…”
Section: B Main Contributionsmentioning
confidence: 90%
“…[4]. The algorithms proposed in this paper, as well as those in [33], [34] and (F)PGM [3], work equally well over the complex numbers C: one only has to change the transpose operation to Hermitian in the appropriate locations. For the same reasons, and in order to simplify the transitions between the PR and ARM framework and PSDMF, we shall use real-valued notations (transpose) also for the PR and ARM equations.…”
Section: Notationsmentioning
confidence: 99%
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“…To obtain upper bounds on the factorization rank of a given matrix one can employ heuristics that try to construct small factorizations. Many such heuristics exist for the nonnegative rank (see the overview [31] and references therein), factorization algorithms exist for completely positive matrices (see the recent paper [40], also [21] for structured completely positive matrices), and algorithms to compute positive semidefinite factorizations are presented in the recent work [75]. In this paper we want to compute lower bounds on matrix factorization ranks, which we achieve by employing a relaxation approach based on (noncommutative) polynomial optimization.…”
mentioning
confidence: 99%