Convex relaxations of structured matrix factorizations

TL;DR

This paper introduces convex relaxations for structured matrix factorizations, providing algorithms and theoretical analysis to compute approximate decompositions with guarantees, especially for factors with specific structures like positivity or sparsity.

Contribution

It establishes the equivalence between optimal matrix factorization and gauge functions, and develops semi-definite relaxations and algorithms with approximation guarantees for structured factorizations.

Findings

Semi-definite relaxations effectively approximate structured matrix factorizations.

Algorithms can recover approximate decompositions with provable guarantees.

Simulations demonstrate successful decomposition with elements in 0,1.

Abstract

We consider the factorization of a rectangular matrix $X $ into a positive linear combination of rank-one factors of the form $u v^\top$, where $u$ and $v$ belongs to certain sets $\mathcal{U}$ and $\mathcal{V}$, that may encode specific structures regarding the factors, such as positivity or sparsity. In this paper, we show that computing the optimal decomposition is equivalent to computing a certain gauge function of $X$ and we provide a detailed analysis of these gauge functions and their polars. Since these gauge functions are typically hard to compute, we present semi-definite relaxations and several algorithms that may recover approximate decompositions with approximation guarantees. We illustrate our results with simulations on finding decompositions with elements in $\{0,1\}$. As side contributions, we present a detailed analysis of variational quadratic representations of norms as well as a new iterative basis pursuit algorithm that can deal with inexact first-order oracles.