Blind Deconvolution Using Convex Programming

Abstract

We consider the problem of recovering two unknown vectors, w and x, of length L from their circular convolution. We make the structural assumption that the two vectors are members known subspaces, one with dimension N and the other with dimension K. Although the observed convolution is nonlinear in both w and x, it is linear in the rank-1 matrix formed by their outer product wx * . This observation allows us to recast the deconvolution problem as low-rank matrix recovery problem from linear measurements, whose natural convex relaxation is a nuclear norm minimization program.We prove the effectiveness of this relaxation by showing that for "generic" signals, the program can deconvolve w and x exactly when the maximum of N and K is almost on the order of L. That is, we show that if x is drawn from a random subspace of dimension N , and w is a vector in a subspace of dimension K whose basis vectors are "spread out" in the frequency domain, then nuclear norm minimization recovers wx * without error. We discuss this result in the context of blind channel estimation in communications. If we have a message of length N which we code using a random L × N coding matrix, and the encoded message travels through an unknown linear time-invariant channel of maximum length K, then the receiver can recover both the channel response and the message when L N + K, to within constant and log factors.

A convex approach to blind deconvolution with diverse inputs

“…In order to show that Y P obtained by golfing satisfies the conditions (8) and (9), we need two auxiliary lemmas. Their proofs will appear in the long version of this note, and are inspired by similar developments in [5]. Lemma 1.…”

“…It was established in [5] (see [2], [8] for important background material) that hm H is the unique minimizer of (2) when there exists a (dual certificate) Y such that…”

“…It is possible to show (see [5], lemma 4) that µ 2 hp ≤ µ 2 h and µ 2 mp ≤ µ 2 m with high probability. Lemma 2.…”

“…See for example [1], [2], [3], [4], [5], [6], [7]. The idea is to lift the original problem into a problem with linear constraints in a higher-dimensional space, and relax any rank condition with a convex substitute.…”

“…Notice that the framework in this paper differs from that in [5] in that the impulse response h is arbitrary, rather than a member of a known subspace. It is the "diversity" afforded by the presence of a small number of random inputs, in contrast to a single random input, that enables us to consider an arbitrary h.…”