Sparse linear regression with compressed and low-precision data via concave quadratic programming

We consider the problem of the recovery of a k-sparse vector from compressed linear measurements when data are corrupted by a quantization noise. When the number of measurements is not sufficiently large, different k-sparse solutions may be present in the feasible set, and the classical ℓ₁ approach may be unsuccessful. For this motivation, we propose a non-convex quadratic programming method, which exploits prior information on the magnitude of the non-zero parameters. This results in a more efficient support recovery. We provide sufficient conditions for successful recovery and numerical simulations to illustrate the practical feasibility of the proposed method.

I. Introduction

Sparse linear regression is the recovery of a sparse vector from linear measurements , with . A vector is sparse if it has few nonzero components; more precisely, we call k-sparse a vector with non-zero components. The interest for sparse solutions has different motivations. In machine learning and system identification, a purpose is to build models as simple as possible from large datasets. Indeed, we know that in many cases the true number of parameters of a system is much smaller than the global dimensionality of the problem, and sparsity supports the removal of redundant parameters. In the recent literature, the identification of linear systems under sparsity constraints is considered in [1], [2], [3], [4]. Furthermore, in the last decade, sparsity has been attracting a lot of attention due to the theory of compressed sensing (CS, [5], [6]), which states that a sparse vector can be recovered from compressed linear measurements, that is, when m < n, under suitable conditions on the structure of A.