Blind calibration for compressed sensing by convex optimization

TL;DR

This paper introduces a convex optimization method for blind calibration in compressed sensing systems with unknown gains, enabling effective calibration using sparse signals and demonstrating sharp phase transition behavior.

Contribution

It reformulates the blind calibration problem as a convex optimization task, making it solvable with standard algorithms, which was previously considered highly non-convex.

Findings

Convex formulation enables exact calibration with off-the-shelf algorithms.

Method is effective even with highly uncalibrated measures.

Success exhibits sharp phase transition behavior.

Abstract

We consider the problem of calibrating a compressed sensing measurement system under the assumption that the decalibration consists in unknown gains on each measure. We focus on {\em blind} calibration, using measures performed on a few unknown (but sparse) signals. A naive formulation of this blind calibration problem, using $\ell_{1}$ minimization, is reminiscent of blind source separation and dictionary learning, which are known to be highly non-convex and riddled with local minima. In the considered context, we show that in fact this formulation can be exactly expressed as a convex optimization problem, and can be solved using off-the-shelf algorithms. Numerical simulations demonstrate the effectiveness of the approach even for highly uncalibrated measures, when a sufficient number of (unknown, but sparse) calibrating signals is provided. We observe that the success/failure of the approach seems to obey sharp phase transitions.