Non-Convex Compressed Sensing with Training Data

TL;DR

This paper introduces a novel approach to solve non-convex compressed sensing problems by leveraging training data, enabling high-probability solutions with minimal assumptions on the measurement matrix.

Contribution

It proposes using training problems within a neural network framework to address NP-hard compressed sensing challenges without requiring RIP conditions.

Findings

High-probability recovery of sparse solutions using training data

Effective in scenarios with minimal assumptions on matrix A

Applicable to under-determined linear systems

Abstract

Efficient algorithms for the sparse solution of under-determined linear systems $Ax = b$ are known for matrices $A$ satisfying suitable assumptions like the restricted isometry property (RIP). Without such assumptions little is known and without any assumptions on $A$ the problem is $NP$-hard. A common approach is to replace $\ell_1$ by $\ell_p$ minimization for $0 < p < 1$, which is no longer convex and typically requires some form of local initial values for provably convergent algorithms. In this paper, we consider an alternative, where instead of suitable initial values we are provided with extra training problems $Ax = B_l$, $l=1, \dots, p$ that are related to our compressed sensing problem. They allow us to find the solution of the original problem $Ax = b$ with high probability in the range of a one layer linear neural network with comparatively few assumptions on the matrix $A$.