Blind Deconvolution: Convex Geometry and Noise Robustness

Blind deconvolution problems arise in many areas in science and engineering such as communications and astronomy. For this reason, this problem has been a subject of intense study for many decades. Recently, motivated by the success of randomization in compressed sensing and low-rank matrix recovery, a new viewpoint has been introduced. Namely one assumes that the convolved signals are contained in known subspaces, which possess a certain degree of randomness. Such a scenario appears, for example, in wireless communications. Here the idea is to randomly embed the signal into a higher dimensional space before transmission through an unknown channel. The resulting redundancy can then be used for recovery. The first approach for this subspace model, proposed by Ahmed, Recht, and Romberg, was to lift the problem into the space of matrix representation and use the nuclear norm as a regularizer. Their recovery guarantees also apply for noisy measurements, but the error bounds involve seemingly suboptimal dimensional scaling factors. In this paper we will introduce a new geometric analysis based on the conic singular value of the descent cone, which explains these factors. Furthermore, we show that for O(1) noise-levels, these factors can be avoided and one can obtain near-optimal error bounds.