Non-Convex Compressed Sensing with Training Data
This addresses a bottleneck in compressed sensing for scenarios where traditional assumptions fail, offering a novel approach with potential applications in signal processing and data recovery.
The paper tackles the problem of solving under-determined linear systems in compressed sensing without strong matrix assumptions by using training data to find solutions via a linear neural network, achieving high-probability success with minimal assumptions.
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 $A$.