Asymptotic of Sparse Support Recovery for Positive Measures

Sparse Spikes Deconvolution on Thin Grids

“…Proof. We define a candidate solution â by âJ = a 0,J + A + J w − λv J , âJ c = 0 and we prove that â is the unique solution to (P λ (y 0 + w)) using the optimality conditions (15) and (16).…”

“…whatever the spacing between the Diracs), η ∞V is always a non-degenerate certificate (in the sense of Proposition 3), meaning that one actually has η ∞ V = η ∞ 0 (where the minimal norm certificate η ∞ 0 is defined in(37)). This empirical finding is the subject of another recent work on the asymptotic of sparse recovery of positive measures when the spacing between the Diracs tends to zero[16]. Since η ∞ 0 is non-degenerate, one can thus apply Theorem 2 to analyze the extended support of the Lasso (see below Section 6.2 for a numerical illustration).For the C-BP problem, the situation is however more contrasted.…”

Multi-kernel Unmixing and Super-Resolution Using the Modified Matrix Pencil Method

“…The resulting minimization problem is an infinite dimensional convex program over Borel measures on R. It was shown that the dual of this problem can be formulated as a semi-definite program (SDP) in finitely many variables, and thus can be solved efficiently. Since then, there have been numerous follow-up works such as by Schiebinger et al [42], Duval and Peyre [14], Denoyelle et al [13], Bendory et al [3], Azaïs et al [2] and many others. For instance, [42] considers the noiseless setting by taking real-valued samples of y with a more general choice of g (such as a Gaussian) and also assumes x to be non-negative.…”

“…Bendory et al [3] consider g to be Gaussian or Cauchy, do not place sign assumptions on x, and also analyze TV norm minimization with linear fidelity constraints for estimating x from noiseless samples of y. The approach adopted in [14,13] is to solve a least-squares-type minimization procedure with a TV norm based penalty term (also referred to as the Beurling LASSO (see for eg., [2])) for recovering x from samples of y. The approach in [15] considers a natural finite approximation on the grid to the continuous problem, and studies the limiting behaviour as the grid becomes finer; see also [16].…”

“…In the noisy setting, the state of affairs is radically different since it is known (see for eg., [10,Section 3.2], [36,Corollary 1.4]) that some separation between the spike locations is indeed necessary for stable recovery. When sufficient separation is assumed and provided the noise level is small enough, then stable recovery of x is possible (see for eg., [18,14,13,2]).…”