Nonconvex Demixing From Bilinear Measurements

Abstract

We consider the problem of demixing a sequence of source signals from the sum of noisy bilinear measurements. It is a generalized mathematical model for blind demixing with blind deconvolution, which is prevalent across the areas of dictionary learning, image processing, and communications. However, state-of-the-art convex methods for blind demixing via semidefinite programming are computationally infeasible for large-scale problems. Although the existing nonconvex algorithms are able to address the scaling issue, they normally require proper regularization to establish optimality guarantees. The additional regularization yields tedious algorithmic parameters and pessimistic convergence rates with conservative step sizes. To address the limitations of existing methods, we thus develop a provable nonconvex demixing procedure via Wirtinger flow, much like vanilla gradient descent, to harness the benefits of regularization free, fast convergence rate with aggressive step size and computational optimality guarantees. This is achieved by exploiting the benign geometry of the blind demixing problem, thereby revealing that Wirtinger flow enforces the regularizationfree iterates in the region of strong convexity and qualified level of smoothness where the step size can be chosen aggressively.Index Terms-Blind demixing, blind deconvolution, bilinear measurements, nonconvex optimization, Wirtinger flow, regularization-free, statistical and computational guarantees.

Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview

“…The incoherence conditions also provably hold across all iterations; see [26] for details. Similar results have been derived for the blind demixing case as well [62].…”

Intelligent Reflecting Surface for Downlink Non-Orthogonal Multiple Access Networks

“…where the inner product is defined as X, Y = R(Tr(X H Y )) according to Wirtinger's calculus [41] in the complex domain and Z denotes Z's feasible solution region. Since the primal problem (27) and its dual problem (28) are still non-convex, the duality-based DC algorithm iteratively updates both the primal and dual variables via successive convex approximation.…”

Blind Over-the-Air Computation and Data Fusion via Provable Wirtinger Flow

“…According to Section IV-A and Lemma 1, the unique challenge in establishing the approximate state evolution (27) is to bound the perturbation terms to certain order, i.e., |ψ h t i |, |ψ x t i |, |ϕ h t i |, |ϕ x t i |, |ρ h t i |, |ρ x t i | ≪ 1/log m for i = 1, · · · , s. To achieve this goal, we exploit some variants of leave-one-out sequences [23], [16] to establish the "nearindependence" between {z t i } and {a i }. Hence, some terms can be approximated by a sum of independent variables with well-controlled weight, thereby be controlled via central limit theorem.…”

“…Recently, blind demixing has become a powerful tool to elude channel-state-information (i.e., without channel estimation at both transmitters and receivers) thereby enabling low-latency communications [13], [14], [15]. Specifically, in blind demixing, a sequence of source signals can be recovered from the sum of bilinear measurements without the knowledge of channel information [16]. Inspired by the recent progress of blind demixing, in this paper, we shall propose a novel blind over-the-air computation (BlairComp) scheme for low-latency data aggregation, thereby computing the desired function (e.g., arithmetic mean) of sensing data vectors without the prior knowledge of channel information.…”