SIGIBE: Solving random bilinear equations via gradient descent with spectral initialization

We investigate the problem of finding the real-valued vectors h, of size L, and x, of size P, from M independent measurements ym = 〈am, h〉〈bm, x〉, where am and bm are known random vectors. Recovery of the unknowns entails solving a set of bilinear equations, a challenging problem encountered in signal processing tasks such as blind deconvolution for channel equalization or image deblurring. Inspired by the Wirtinger flow approach to the related phase retrieval problem, we propose a solver that proceeds in two steps: (i) first a spectral method is used to obtain an initial guess; which is then (ii) refined using simple and scalable gradient descent iterations to minimize a natural non-convex formulation of the recovery problem. Our method - which we refer to as SIGIBE: Spectral Initialization and Gradient Iterations for Bilinear Equations - can accommodate arbitrary correlations between am and bm. Different from recent approaches to blind deconvolution using convex relaxation, SIGIBE does not require matrix lifting that could hinder the method's scalability. Numerical tests corroborate SIGIBE's effectiveness in various data settings, and show successful recovery with as few as M ≳ (L + P) measurements.