Robustness of Sparse Recovery via -Minimization: A Topological Viewpoint

A recent trend in compressed sensing is to consider nonconvex optimization techniques for sparse recovery. The important case of F-minimization has become of particular interest, for which the exact reconstruction condition (ERC) in the noiseless setting can be precisely characterized by the null space property (NSP). However, little work has been done concerning its robust reconstruction condition (RRC) in the noisy setting. We look at the null space of the measurement matrix as a point on the Grassmann manifold, and then study the relation between the ERC and RRC sets, denoted as Ω_J and Ω_J^r, respectively. It is shown that Ω_J is the interior of Ω_J, from which a previous result of the equivalence of ERC and RRC for ℓ_p-minimization follows easily as a special case. Moreover, when F is nondecreasing, it is shown that Ω̅_Jint(Ω_J) is a set of measure zero and of the first category. As a consequence, the probabilities of ERC and RRC are the same if the measurement matrix A is randomly generated according to a continuous distribution. Quantitatively, if the null space N(A) lies in the d-interior of Ω_J, then RRC will be satisfied with the robustness constant C=2+2d/dσ_min(A^T); and conversely, if RRC holds with C=2-2d/dσ_max(A^T), then N(A) must lie in d-interior of Ω_J. We also present several rules for comparing the performances of different cost functions. Finally, these results are capitalized to derive achievable tradeoffs between the measurement rate and robustness with the aid of Gordon´s escape through the mesh theorem or a connection between NSP and the restricted eigenvalue condition.