ICML 2016poster462 citations
Low-rank Solutions of Linear Matrix Equations via Procrustes Flow
Stephen Tu, Ross Boczar, Max Simchowitz, Mahdi Soltanolkotabi, Ben Recht
Abstract
In this paper we study the problem of recovering a low-rank matrix from linear measurements. Our algorithm, which we call Procrustes Flow, starts from an initial estimate obtained by a thresholding scheme followed by gradient descent on a non-convex objective. We show that as long as the measurements obey a standard restricted isometry property, our algorithm converges to the unknown matrix at a geometric rate. In the case of Gaussian measurements, such convergence occurs for a n1 \times n2 matrix of rank r when the number of measurements exceeds a constant times (n1 + n2)r.
BibTeX
@InProceedings{pmlr-v48-tu16,
title = {Low-rank Solutions of Linear Matrix Equations via Procrustes Flow},
author = {Tu, Stephen and Boczar, Ross and Simchowitz, Max and Soltanolkotabi, Mahdi and Recht, Ben},
booktitle = {Proceedings of The 33rd International Conference on Machine Learning},
pages = {964--973},
year = {2016},
editor = {Balcan, Maria Florina and Weinberger, Kilian Q.},
volume = {48},
series = {Proceedings of Machine Learning Research},
address = {New York, New York, USA},
month = {20--22 Jun},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v48/tu16.pdf},
url = {https://proceedings.mlr.press/v48/tu16.html},
abstract = {In this paper we study the problem of recovering a low-rank matrix from linear measurements. Our algorithm, which we call Procrustes Flow, starts from an initial estimate obtained by a thresholding scheme followed by gradient descent on a non-convex objective. We show that as long as the measurements obey a standard restricted isometry property, our algorithm converges to the unknown matrix at a geometric rate. In the case of Gaussian measurements, such convergence occurs for a n1 \times n2 matrix of rank r when the number of measurements exceeds a constant times (n1 + n2)r.}
}