An unbiased approach to compressed sensing

TL;DR

This paper introduces a non-convex functional for compressed sensing that guarantees finding the oracle solution with fewer assumptions on the measurement matrix and no spurious local minima, improving accuracy and reliability.

Contribution

It presents a novel non-convex approach that ensures global optimality of the oracle solution in compressed sensing, even with noise and limited prior information.

Findings

Global minimum corresponds to oracle solution

Lower bounds on estimation error constants

No spurious local minima in the proposed functional

Abstract

In compressed sensing a sparse vector is approximately retrieved from an under-determined equation system $Ax=b$. Exact retrieval would mean solving a large combinatorial problem which is well known to be NP-hard. For $b$ of the form $Ax_0+\epsilon$ where $x_0$ and $\epsilon$ is noise, the `oracle solution' is the one you get if you a priori know the support of $x_0$, and is the best solution one could hope for. We provide a non-convex functional whose global minimum is the oracle solution, with the property that any other local minimizer necessarily has high cardinality. We provide estimates of the type $\|\hat x-x_0\|_2\leq C\|\epsilon\|_2$ with constants $C$ that are significantly lower than for competing methods or theorems, and our theory relies on soft assumptions on the matrix $A$, in comparison with standard results in the field. The framework also allows to incorporate a priori information on the cardinality of the sought vector. In this case we show that despite being non-convex, our cost functional has no spurious local minima and the global minima is again the `oracle solution', thereby providing the first method which is guaranteed to find this point for reasonable levels of noise, without resorting to combinatorial methods.