Polar Deconvolution of Mixed Signals

TL;DR

This paper introduces a polar deconvolution method for separating mixed signals using convex optimization, providing probabilistic error bounds and an efficient algorithm, with theoretical guarantees and confirmed by experiments.

Contribution

It develops a novel two-stage convex approach leveraging polar convolution theory for stable signal demixing with provable error bounds and an efficient solution algorithm.

Findings

High-probability stable recovery of signals under random measurements and bounded noise.

Near-optimal sample complexity for low-complexity, incoherent signals.

Numerical experiments validate theoretical guarantees and algorithm efficiency.

Abstract

The signal demixing problem seeks to separate a superposition of multiple signals into its constituent components. This paper studies a two-stage approach that first decompresses and subsequently deconvolves the noisy and undersampled observations of the superposition using two convex programs. Probabilistic error bounds are given on the accuracy with which this process approximates the individual signals. The theory of polar convolution of convex sets and gauge functions plays a central role in the analysis and solution process. If the measurements are random and the noise is bounded, this approach stably recovers low-complexity and mutually incoherent signals, with high probability and with near-optimal sample complexity. We develop an efficient algorithm, based on level-set and conditional-gradient methods, that solves the convex optimization problems with sublinear iteration complexity and linear space requirements. Numerical experiments on both real and synthetic data confirm the theory and the efficiency of the approach.