Paper Info
Title
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 \(\{A^1,\ldots ,A^m\}\) and \(\{B^1,\ldots ,B^n\}\) such that \(X_{i,j}=\text {trace}(A^iB^j)\) for \(i=1,\ldots ,m\), and \(j=1,\ldots ,n\). PSD factorization is NP-hard. In this work, we introduce several local optimization schemes to tackle this problem: a fast projected gradient method and two algorithms based on the coordinate descent framework. The main application of PSD factorization is the computation of semidefinite extensions, that is, the representations of polyhedrons as projections of spectrahedra, for which the matrix to be factorized is the slack matrix of the polyhedron. We compare the performance of our algorithms on this class of problems. In particular, we compute the PSD extensions of size \(k=1+ \lceil \log _2(n) \rceil \) for the regular n-gons when \(n=5\), 8 and 10. We also show how to generalize our algorithms to compute the square root rank (which is the size of the factors in a PSD factorization where all factor matrices \(A^i\) and \(B^j\) have rank one) and completely PSD factorizations (which is the special case where the input matrix is symmetric and equality \(A^i=B^i\) is required for all i).
Year
DOI
Venue
2018
10.1007/s10589-018-9998-x
Comp. Opt. and Appl.
Keywords
DocType
Volume
Positive semidefinite factorization, Extended formulations, Fast gradient method, Coordinate descent method
Journal
abs/1707.07953
Issue
ISSN
Citations 
1
Computational Optimization and Applications 71 (1), pp. 193-219, 2018
1
PageRank 
References 
Authors
0.35
14
3
Name
Order
Citations
PageRank
A. Vandaele1203.89
francois glineur2132.36
Nicolas Gillis350339.77