Ambiguity and sensitivity in imprecise dictionaries for compressed sensing

Abstract

An imprecise dictionary is commonly used in compressed sensing in cases where the dictionary is derived via a learning process or is based on incomplete prior knowledge. This paper investigates the issue of ambiguity in dictionary estimation and the sensitivity of a true sparse vector and signal (i.e., signals represented by true sparse vectors with respect to the true dictionary) to dictionary perturbations in compressed sensing. We first demonstrate that the inherent ambiguity in dictionary estimation cannot be resolved, even when the desired sparse vector can be obtained. By imposing conditions on the perturbation of the true dictionary, we then demonstrate that the sparse vector and signal can both be stably estimated within the bounds of the dictionary perturbation. Our analysis results are based on the condition that AD holds the restricted isometry property (RIP), where A is a precise sensing matrix and D is a dictionary mismatch. The RIP of AD cannot be efficiently determined for a general over-complete dictionary; therefore, we numerically verified our analytical results by conducting experiments under the specific but commonly encountered situation in which D (an orthonormal matrix) and A (a Gaussian sensing matrix), in which the RIP of AD holds with a high degree of probability.