---
abstract: 'Expressing a matrix as the sum of a low-rank matrix plus a sparse matrix is a flexible model capturing global and local features in data. This model is the foundation of robust principle component analysis [@Candes2011robust; @Chandrasekaran2009ranksparsity], and popularized by dynamic-foreground/static-background separation [@Bouwmans2016decomposition] amongst other applications. Compressed sensing, matrix completion, and their variants [@Eldar2012compressed; @Foucart2013a] have established that data satisfying low complexity models can be efficiently measured and recovered from a number of measurements proportional to the model complexity rather than the ambient dimension. This manuscript develops similar guarantees showing that $m\times n$ matrices that can be expressed as the sum of a rank-$r$ matrix and a $s$-sparse matrix can be recovered by computationally tractable methods from $\mathcal{O}(r(m+n-r)+s)\log(mn/s)$ linear measurements. More specifically, we establish that the restricted isometry constants for the aforementioned matrices remain bounded independent of problem size provided $p/mn$, $s/p$, and $r(m+n-r)/p$ reman fixed. Additionally, we show that semidefinite programming and two hard threshold gradient descent algorithms, NIHT and NAHT, converge to the measured matrix provided the measurement operator’s RIC’s are sufficiently small. Numerical experiments illustrating these results are shown for synthetic problems, dynamic-foreground/static-background separation, and multispectral imaging.'
address:
- 'Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK'
- 'The Alan Turing Institute, The British Library, London NW1 2DB, UK'
author:
- Jared Tanner
- Simon Vary
bibliography:
- 'library.bib'
title: 'Compressed sensing of low-rank plus sparse matrices'
---

matrix sensing ,low-rank plus sparse matrix ,restricted isometry property,non-convex methods ,robust PCA 15A29,41A29,62H25 ,65F10 ,65J20 ,68Q25 ,90C22 ,90C26

Acknowledgement {#acknowledgement .unnumbered}
===============

We would like to thank Robert A. Lamb and David Humphreys for useful discussions around the applications of low-rank plus sparse model to multispectral imaging.
