On a unified view of nullspace-type conditions for recoveries associated with general sparsity structures

Abstract

We discuss a general notion of "sparsity structure" and associated recoveries of a sparse signal from its linear image of reduced dimension possibly corrupted with noise. Our approach allows for unified treatment of (a) the "usual sparsity" and "usual ℓ 1 recovery," (b) block-sparsity with possibly overlapping blocks and associated block-ℓ 1 recovery, and (c) low-rank-oriented recovery by nuclear norm minimization. The proposed recovery routines are natural extensions of the usual ℓ 1 minimization used in Compressed Sensing. Specifically, we present nullspace-type sufficient conditions for the recovery to be precise on sparse signals in the noiseless case. Then we derive error bounds for imperfect (nearly sparse signal, presence of observation noise, etc.) recovery under these conditions. In all of these cases, we present efficiently verifiable sufficient conditions for the validity of the associated nullspace properties.

Computational complexity versus statistical performance on sparse recovery problems

“…It is not clear how this view can be used for latent overlapping group norms presented by Obozinski et al [2011] applied in biology. Moreover the sufficient conditions that Juditsky et al [2014] present are sufficient but not necessary in the nuclear norm case. Better characterizing the key geometrical properties of these norms is therefore a challenging research direction.…”

“…Moreover, it is not clear if their definition is sufficient to characterize the conic nature of these norms, in particular in the nuclear norm case that will require additional linear algebra results. In comparison, the framework proposed by Juditsky et al [2014] can encompass non-latent overlapping groups. For future use, we simplify the third property of their definition [Juditsky et al, 2014, Section 2.1] in Appendix B.…”

“…We quickly discuss the framework of Juditsky et al [2014] for sparsity inducing norms and show that it can be simplified. We first recall the definition.…”

Sparse recovery by reduced variance stochastic approximation

“…In what follows we assume to be given a sparsity structure [27] on E-a family P of projector mappings P = P 2 on E with associated nonnegative weights π(P ). For a nonnegative real s we set P s = {P ∈ P : π(P ) ≤ s}.…”

A perturbation inequality for concave functions of singular values and its applications in low-rank matrix recovery

“…A first study in this direction was done by Recht, Fazel and Parrilo [29], who showed that techniques used to analyze the ℓ 1 heuristic for sparse vector recovery (see [33] for an overview and further pointers to the literature) can be extended to analyze the nuclear norm heuristic. Since then, recovery conditions based on the restricted isometry property (RIP) and various nullspace properties have been established for the nuclear norm heuristic; see, e.g., [28,5,4,18] for some recent results. In fact, many recovery conditions for the nuclear norm heuristic can be derived in a rather simple fashion from their counterparts for the ℓ 1 heuristic by utilizing a perturbation inequality for the nuclear norm [28].…”