# Painless Breakups -- Efficient Demixing of Low Rank Matrices

## TL;DR

This paper introduces two efficient algorithms for demixing multiple low-rank matrices from combined measurements, with theoretical guarantees and applications in quantum tomography and IoT.

## Contribution

The paper proposes novel hard thresholding algorithms for low rank matrix demixing with proven linear convergence guarantees.

## Key findings

- Algorithms successfully demix low-rank matrices in simulations.
- Theoretical analysis confirms linear convergence under certain conditions.
- Applications demonstrated in quantum tomography and IoT contexts.

## Abstract

Assume we are given a sum of linear measurements of $s$ different rank-$r$ matrices of the form $y = \sum_{k=1}^{s} \mathcal{A}_k ({X}_k)$. When and under which conditions is it possible to extract (demix) the individual matrices ${X}_k$ from the single measurement vector ${y}$? And can we do the demixing numerically efficiently? We present two computationally efficient algorithms based on hard thresholding to solve this low rank demixing problem. We prove that under suitable conditions these algorithms are guaranteed to converge to the correct solution at a linear rate. We discuss applications in connection with quantum tomography and the Internet-of-Things. Numerical simulations demonstrate empirically the performance of the proposed algorithms.