A Tighter Bound on the Uniqueness of Sparsely-Used Dictionaries

In this paper, we consider the sparsely used dictionary learning problem and focus on the case that the dictionary is square with arbitrary entries and the coefficient matrix is a sparse with random entries, which is first proposed and studied by Wang and Spielman et al.[1], [2]. We improve the theoretical results with a tighter bounded uniqueness theorem, which says that O(n) samples are sufficient to uniquely determine the decomposition with probability one when given assumption is satisfied, and the proof is given.