Papers › Unique sparse decomposition of low rank matrices
Unique sparse decomposition of low rank matrices
Dian Jin, Xin Bing, Yuqian Zhang
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∈ℝ^(p×n) that admits a sparse representation. Specifically, we consider Y = A X∈ℝ^(p×n) where the matrix A∈ℝ^(p×r) has full column rank, with r < min{n,p}, and the matrix X∈ℝ^(r×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.
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