Blind Demixing and Deconvolution at Near-Optimal Rate

Abstract

We consider simultaneous blind deconvolution of r source signals from their noisy superposition, a problem also referred to blind demixing and deconvolution. This signal processing problem occurs in the context of the Internet of Things where a massive number of sensors sporadically communicate only short messages over unknown channels. We show that robust recovery of message and channel vectors can be achieved via convex optimization when random linear encoding using i.i.d. complex Gaussian matrices is used at the devices and the number of required measurements at the receiver scales with the degrees of freedom of the overall estimation problem. Since the scaling is linear in r our result significantly improves over recent works.

Nonconvex Demixing From Bilinear Measurements

“…The extension turns out to be nontrivial since the "incoherence" between multiple sources for blind demixing leads to distortion to the statistical property in the single source scenario for blind deconvolution. The similar challenge has also been observed in [1], [2] by extending the convex relaxation approach (i.e., semidefinite programming) for blind deconvolution to the setting of blind demixing. Furthermore, the noisy measurements also bring additional challenges to establish theoretical guarantees.…”

“…The incoherence between b j and h i for 1 ≤ i ≤ s, 1 ≤ j ≤ m specifies the smoothness of the loss function (2). Within the region of incoherence and contraction (defined in Section IV-A) that enjoys the qualified level of smoothness, the step size for iterative refinement procedure can be chosen more aggressively according to generic optimization theory [10].…”

“…Demixing a sequence of source signals from the sum of bilinear measurements provides a generalized mathematical modeling framework for blind demixing with blind deconvolution [1], [2], [3]. It spans a wide scope of applications ranging from communication [4], imaging [5], and machine learning [6], to the recent application in the context of the Internetof-Things for sporadic and short messages communications over unknown channels [3].…”

Regularized gradient descent: a non-convex recipe for fast joint blind deconvolution and demixing

“…Our previous work [23] shows that the convex approach via semidefinite programming (see (2.10)) requires L ≥ C 0 s 2 (K + µ 2 h N ) log 3 (L) to ensure exact recovery. Later, [15] improves this result to the near-optimal bound L ≥ C 0 s(K + µ 2 h N ) up to some log-factors. The difference between nonconvex and convex methods lies in the appearance of the condition number κ in (3.5).…”

“…The question of when the solution of (2.10) yields exact recovery is first answered in our previous work [23]. Late, [29,15] have improved this result to the near-optimal bound L ≥ C 0 s(K + N ) up to some log-factors where the main theoretical result is informally summarized in the following theorem. While the SDP relaxation is definitely effective and has theoretic performance guarantees, the computational costs for solving an SDP already become too expensive for moderate size problems, let alone for large scale problems.…”