Towards Designing Optimal Sensing Matrices for Generalized Linear Inverse Problems

We consider an inverse problem $\mathbf{y}= f(\mathbf{Ax})$, where $\mathbf{x}\in\mathbb{R}^n$ is the signal of interest, $\mathbf{A}$ is the sensing matrix, $f$ is a nonlinear function and $\mathbf{y} \in \mathbb{R}^m$ is the measurement vector. In many applications, we have some level of freedom to design the sensing matrix $\mathbf{A}$, and in such circumstances we could optimize $\mathbf{A}$ 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 $\mathbf{x}$ from $\mathbf{y}$. 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 \bmmathbfA} 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.