Unique Sparse Decomposition of Low Rank Matrices

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