Towards Designing Optimal Sensing Matrices for Generalized Linear Inverse Problems

Abstract

We consider an inverse problem y= f(Ax), where xR^n is the signal of interest, A is the sensing matrix, f is a nonlinear function and y R^m is the measurement vector. In many applications, we have some level of freedom to design the sensing matrix A, and in such circumstances we could optimize A to achieve better reconstruction performance. As a first step towards optimal design, it is important to understand the impact of the sensing matrix on the difficulty of recovering x from y. In this paper, we study the performance of one of the most successful recovery methods, i.e., the expectation propagation (EP) algorithm. We define a notion of spikiness for the spectrum of and show the importance of this measure for the performance of EP. We show that whether a spikier spectrum can hurt or help the recovery performance depends on f. Based on our framework, we are able to show that, in phase-retrieval problems, matrices with spikier spectrums are better for EP, while in 1-bit compressed sensing problems, less spiky spectrums lead to better performance. Our results unify and substantially generalize existing results that compare Gaussian and orthogonal matrices, and provide a platform towards designing optimal sensing systems.

Tasks

Reproductions