  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.
