Self-Calibration and Biconvex Compressive Sensing

TL;DR

This paper introduces a novel method called SparseLift that enables automatic self-calibration of sensors by framing the problem as a biconvex compressive sensing task, allowing for exact and robust recovery of calibration errors and signals.

Contribution

It develops a new convex optimization framework for self-calibration using biconvex compressive sensing and provides theoretical guarantees for exact recovery.

Findings

Exact recovery of calibration parameters and signals under certain conditions

Robustness of the method against noise and model errors

Numerical simulations confirm theoretical results

Abstract

The design of high-precision sensing devises becomes ever more difficult and expensive. At the same time, the need for precise calibration of these devices (ranging from tiny sensors to space telescopes) manifests itself as a major roadblock in many scientific and technological endeavors. To achieve optimal performance of advanced high-performance sensors one must carefully calibrate them, which is often difficult or even impossible to do in practice. In this work we bring together three seemingly unrelated concepts, namely Self-Calibration, Compressive Sensing, and Biconvex Optimization. The idea behind self-calibration is to equip a hardware device with a smart algorithm that can compensate automatically for the lack of calibration. We show how several self-calibration problems can be treated efficiently within the framework of biconvex compressive sensing via a new method called SparseLift. More specifically, we consider a linear system of equations y = DAx, where both x and the diagonal matrix D (which models the calibration error) are unknown. By "lifting" this biconvex inverse problem we arrive at a convex optimization problem. By exploiting sparsity in the signal model, we derive explicit theoretical guarantees under which both x and D can be recovered exactly, robustly, and numerically efficiently via linear programming. Applications in array calibration and wireless communications are discussed and numerical simulations are presented, confirming and complementing our theoretical analysis.