Multichannel Blind Deconvolution Using Low Rank Recovery

We introduce a new algorithm for multichannel blind deconvolution. Given the outputs of K linear time-invariant channels driven by a common source, we wish to recover their impulse responses without knowledge of the source signal. Abstractly, this problem amounts to finding a solution to an overdetermined system of quadratic equations. We show how we can recast the problem as solving a system of underdetermined linear equations with a rank constraint. Recent results in the area of low rank recovery have shown that there are effective convex relaxations to problems of this type that are also scalable computationally, allowing us to recover 100s of channel responses after a moderate observation time. We illustrate the effectiveness of our methodology with a numerical simulation of a passive "noise imaging" experiment.