Unique sparse decomposition of low rank matricesDownload PDF

21 May 2021, 20:42 (modified: 21 Jan 2022, 16:31)NeurIPS 2021 PosterReaders: Everyone
Keywords: matrix factorization, low-rank decomposition, nonconvex optimization, second-order geometry, sparse representation, unsupervised learning
TL;DR: 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 by recovering ground-truth $A$ column by column, 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
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Code: https://github.com/Jindiande/Unique_Fac_of_Low_Rank
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