Multiscale Blind Source Separation

TL;DR

This paper introduces SLAM, a multiscale method for blind source separation of piecewise constant signals with unknown mixing weights, achieving exact recovery and confidence sets under finite alphabet assumptions, with applications in genetics.

Contribution

The paper develops a novel multiscale approach for blind source separation with finite alphabet sources, providing exact recovery, confidence sets, and efficient algorithms.

Findings

Exact recovery within an epsilon neighborhood for finite alphabet sources.

Uniform confidence sets and near-optimal rates for Gaussian errors.

Successful application to genetic sequencing data for copy-number analysis.

Abstract

We provide a new methodology for statistical recovery of single linear mixtures of piecewise constant signals (sources) with unknown mixing weights and change points in a multiscale fashion. We show exact recovery within an $\epsilon$-neighborhood of the mixture when the sources take only values in a known finite alphabet. Based on this we provide the SLAM (Separates Linear Alphabet Mixtures) estimators for the mixing weights and sources. For Gaussian error, we obtain uniform confidence sets and optimal rates (up to log-factors) for all quantities. SLAM is efficiently computed as a nonconvex optimization problem by a dynamic program tailored to the finite alphabet assumption. Its performance is investigated in a simulation study. Finally, it is applied to assign copy-number aberrations from genetic sequencing data to different clones and to estimate their proportions.