Unique sparse decomposition of low rank matrices

TL;DR

This paper establishes conditions under which a low rank matrix with a sparse factorization can be uniquely decomposed, and proposes a nonconvex optimization approach with guarantees for recovering the true factors.

Contribution

It provides a theoretical framework for the unique sparse decomposition of low rank matrices and introduces a nonconvex optimization method with provable recovery guarantees.

Findings

Unique sparse decomposition is achievable under certain conditions.

Any strict local solution of the nonconvex problem is close to the true solution.

Numerical experiments validate the theoretical results.

Abstract

The problem of finding the unique low dimensional decomposition of a given matrix has been a fundamental and recurrent problem in many areas. In this paper, we study the problem of seeking a unique decomposition of a low rank matrix $Y\in \mathbb{R}^{p\times n}$ that admits a sparse representation. Specifically, we consider $Y = A X\in \mathbb{R}^{p\times n}$ where the matrix $A\in \mathbb{R}^{p\times r}$ has full column rank, with $r < \min\{n,p\}$, and the matrix $X\in \mathbb{R}^{r\times n}$ is element-wise sparse. We prove that this sparse decomposition of $Y$ can be uniquely identified, up to some intrinsic signed permutation. Our approach relies on solving a nonconvex optimization problem constrained over the unit sphere. Our geometric analysis for the nonconvex optimization landscape shows that any {\em strict} local solution is close to the ground truth solution, and can be recovered by a simple data-driven initialization followed with any second order descent algorithm. At last, we corroborate these theoretical results with numerical experiments.