  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.
