Deep generative demixing: Recovering Lipschitz signals from noisy subgaussian mixtures

TL;DR

This paper establishes theoretical bounds for recovering two Lipschitz signals, including those generated by neural networks, from noisy subgaussian mixtures, extending compressed sensing results to more complex, non-convex structures.

Contribution

It generalizes sample complexity bounds for demixing Lipschitz signals, including non-convex structures, from Gaussian to subgaussian measurement models, with empirical validation using GNNs.

Findings

Sample complexity bounds for Lipschitz signals recovery.

Extension of compressed sensing results to subgaussian measurements.

Numerical simulations demonstrating GNN-based demixing efficacy.

Abstract

Generative neural networks (GNNs) have gained renown for efficaciously capturing intrinsic low-dimensional structure in natural images. Here, we investigate the subgaussian demixing problem for two Lipschitz signals, with GNN demixing as a special case. In demixing, one seeks identification of two signals given their sum and prior structural information. Here, we assume each signal lies in the range of a Lipschitz function, which includes many popular GNNs as a special case. We prove a sample complexity bound for nearly optimal recovery error that extends a recent result of Bora, et al. (2017) from the compressed sensing setting with gaussian matrices to demixing with subgaussian ones. Under a linear signal model in which the signals lie in convex sets, McCoy & Tropp (2014) have characterized the sample complexity for identification under subgaussian mixing. In the present setting, the signal structure need not be convex. For example, our result applies to a domain that is a non-convex union of convex cones. We support the efficacy of this demixing model with numerical simulations using trained GNNs, suggesting an algorithm that would be an interesting object of further theoretical study.