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.

翻译：找到特定矩阵独特的低维分解问题一直是许多领域一个根本性和反复出现的问题。 在本文中, 我们研究的是寻找一种独特的分解问题, 低级矩阵 $Y\ in\ mathbb{R ⁇ p\timen} n} 美元, 允许代表量少。 具体地说, 我们考虑的是 $Y = A X\in \ mathbb{R ⁇ p\times n} 美元, 其中矩阵 $A\ in\ mathbb{R ⁇ p\time r} 问题, 这个问题在许多领域是一个根本性和反复出现的问题。 我们研究的是, 寻找一种独特的分解问题, 美元 = A X\ in a X = a X = A X = A X = A X = A X n_in = A X n mathb{R\ {R\\\ timems n} $ 。 我们的方法依赖于解决一个非convex 优化问题的方式。 我们对非convex press press press press press brationalendalismission exmalation exmessolvealation.