Nonconvex Demixing From Bilinear Measurements

TL;DR

This paper introduces a fast, regularization-free nonconvex algorithm based on Wirtinger flow for demixing signals from bilinear measurements, overcoming scalability issues of previous convex and nonconvex methods.

Contribution

It develops a provable Wirtinger flow algorithm that exploits problem geometry for efficient, regularization-free demixing, with strong convergence guarantees and large step sizes.

Findings

Achieves computational optimality for large-scale demixing problems.

Provides convergence guarantees without regularization.

Enables aggressive step sizes due to benign problem geometry.

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 viaWirtinger 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 regularization-free iterates in the region of strong convexity and qualified level of smoothness, where the step size can be chosen aggressively.