Algorithms for Positive Semidefinite Factorization

TL;DR

This paper introduces local optimization algorithms for positive semidefinite factorization, a complex problem with applications in semidefinite extensions and polyhedral representations, demonstrating their effectiveness on specific geometric cases.

Contribution

The paper presents novel local optimization schemes, including a projected gradient method and coordinate descent algorithms, for tackling the NP-hard PSD factorization problem.

Findings

Successfully computed PSD extensions for regular polygons with sizes 5, 8, and 10.

Demonstrated algorithms' ability to approximate square root rank and completely PSD factorizations.

Compared performance of algorithms on semidefinite extension problems.

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 $X_{i,j}=\text{trace}(A^iB^j)$ for $i=1,...,m$, and $j=1,...n$. 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 $i$).