Journal of Fourier Analysis and Applications

, Volume 25, Issue 1, pp 1–31 | Cite as

Painless Breakups—Efficient Demixing of Low Rank Matrices

  • Thomas Strohmer
  • Ke WeiEmail author


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.


Demixing Low-rank matrices Random matrices Nonconvex optimization Iterative hard thresholding Quantum tomography Restricted isometry property Blind deconvolution 

Mathematics Subject Classification

15A29 15A52 41A29 65F22 90C26 93C41 


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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California at DavisDavisUSA

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