Exact Recovery Conditions for Sparse Representations With Partial Support Information

Abstract

To cite this version:Cédric Herzet, Charles Soussen, Jérôme Idier, Rémi Gribonval. Abstract-We address the exact recovery of a k-sparse vector in the noiseless setting when some partial information on the support is available. This partial information takes the form of either a subset of the true support or an approximate subset including wrong atoms as well. We derive a new sufficient and worst-case necessary (in some sense) condition for the success of some procedures based on ℓp-relaxation, Orthogonal Matching Pursuit (OMP) and Orthogonal Least Squares (OLS). Our result is based on the coherence µ of the dictionary and relaxes the wellknown condition µ < 1/(2k − 1) ensuring the recovery of any k-sparse vector in the non-informed setup. It reads µ < 1/(2k − g + b − 1) when the informed support is composed of g good atoms and b wrong atoms. We emphasize that our condition is complementary to some restrictedisometry based conditions by showing that none of them implies the other.Because this mutual coherence condition is common to all procedures, we carry out a finer analysis based on the Null Space Property (NSP) and the Exact Recovery Condition (ERC). Connections are established regarding the characterization of ℓp-relaxation procedures and OMP in the informed setup. First, we emphasize that the truncated NSP enjoys an ordering property when p is decreased. Second, the partial ERC for OMP (ERC-OMP) implies in turn the truncated NSP for the informed ℓ 1 problem, and the truncated NSP for p < 1.

Nearly Optimal Bounds for Orthogonal Least Squares

“…Consequently, the OLS and OMP algorithms coincide for the first iteration but usually differ afterward (see Section II-A for the justification). It has been empirically observed that OLS is computationally more expensive yet is more reliable than OMP [4]. For more details on the differences between these two algorithms, see [8] and the references therein.…”

A theoretical perspective of solving phaseless compressive sensing via its nonconvex relaxation

“…In the past decade, Compressive Sensing (CS) has gained intensive attention, see [5,6,10,16,24] and references therein. It aims at recovering a sparsest vector from an underdetermined system of linear equations.…”

$NP/CMP$ Equivalence: A Phenomenon Hidden Among Sparsity Models $l_{0}$ Minimization and $l_{p}$ Minimization for Information Processing

“…Unfortunately, although the l 0 -norm characterizes the sparsity of the vector x, the optimization problem (P 0 ) is actually NP-Hard because of the discrete and discontinuous nature of the l 0 -norm. This has resulted in many substitution models for (P 0 ), where x 0 is replaced with functions that evaluate the desirability of a would-be solution to Ax = b (see, e.g., [4], [11], [17], [20], [26], and references therein). Because of the relationship…”

“…A lot of excellent theoretical work (see, e.g., [4], [10], [12], [19]), together with some empirical evidence (see, e.g., [6]), has shown that, provided some conditions are met, such as assuming the restricted isometric property (RIP), the l 1 -norm minimization (P 1 ) can really make an exact recovery. The original notion of RIP has received much attention and has already been tailored to a more general case where 0 < p < 1 (see, e.g., [7], [9], [20]). Work undertaken by Donoho and Tanner in [11] using convex geometry demonstrated a surprising phenomenon that for any real matrix A, whenever the nonnegative solution to (P 0 ) is sufficiently sparse, it is also a unique solution to (P 1 ).…”