Computing and analyzing recoverable supports for sparse reconstruction

Abstract

Designing computational experiments involving ℓ 1 minimization with linear constraints in a finite-dimensional, real-valued space for receiving a sparse solution with a precise number k of nonzero entries is, in general, difficult. Several conditions were introduced which guarantee that, for example for small k or for certain matrices, simply placing entries with desired characteristics on a randomly chosen support will produce vectors which can be recovered by ℓ 1 minimization. In this work, we consider the case of large k and introduce a method which constructs vectors which support has the cardinality k and which can be recovered via ℓ 1 minimization. Especially, such vectors with largest possible support can be constructed. Further, we propose a methodology to quickly check whether a given vector is recoverable. This method can be cast as a linear program and we compare it with solving ℓ 1 minimization directly. Moreover, we gain new insights in the recoverability in a non-asymptotic regime. Our proposal for quickly checking vectors bases on optimality conditions for exact solutions of the ℓ 1 minimization. These conditions can be used to establish equivalence classes of recoverable vectors which have a support of the same cardinality. Further, by these conditions we deduce a geometrical interpretation which identifies an equivalence class with a face of an hypercube which is cut by a certain affine subspace. Due to the new geometrical interpretation we derive new results on the number of equivalence classes which are illustrated by computational experiments.