Blind Deconvolution by a Steepest Descent Algorithm on a Quotient Manifold

Abstract

In this paper, we propose a Riemannian steepest descent method for solving a blind deconvolution problem, which is to recover two unknown signals from their circular convolution. We assume that the two signals are in two known subspaces, one of which is assumed random Gaussian and the other is given by a basis that satisfies a low coherence condition. We prove that the proposed algorithm with an appropriate initialization will recover the exact solution with high probability when the number of measurements is, up to log-factors, the information-theoretical minimum scaling. The quotient structure in our formulation yields a simpler penalty term in the cost function compared to [X. Li et al., Appl. Comput. Harmon. Anal., to appear], which eases the convergence analysis and yields a natural implementation. Empirically, the proposed algorithm has better performance than the Wirtinger gradient descent algorithm and an alternating minimization algorithm in the sense that (i) it needs fewer operations, such as DFTs and matrix-vector multiplications, to reach a similar accuracy, and (ii) it has a higher probability of successful recovery in synthetic tests. An image deblurring problem is also used to demonstrate the efficiency and effectiveness of the proposed algorithm.