Painless Breakups - Efficient Demixing of Low Rank Matrices.

Assume we are given a sum of linear measurements of s different rank-r matrices of the form \(\varvec{y}= \sum _{k=1}^{s} \mathcal {A}_k (\varvec{X}_k)\). When and under which conditions is it possible to extract (demix) the individual matrices \(\varvec{X}_k\) from the single measurement vector \(\varvec{y}\)? 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 introduce an Amalgam-Restricted Isometry Property which is especially suitable for demixing problems and prove that under appropriate 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 the empirical performance of the proposed algorithms.