Unique sparse decomposition of low rank matrices

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 $Y=AX\in {\mathbb{R}}^{p\times n}$ 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.

Accepted by 2021 Neurips, in IEEE Transactions on Information Theory, 2022

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• Generalized power method for sparse principal component analysis
• From Symmetry to Geometry: Tractable Nonconvex Problems
• Efficient Sparse Coding using Hierarchical Riemannian Pursuit