Sketchy Decisions: Convex Low-Rank Matrix Optimization with Optimal Storage
Alp Yurtsever, Madeleine Udell, Joel Tropp, Volkan Cevher
Abstract
This paper concerns a fundamental class of convex matrix optimization problems. It presents the first algorithm that uses optimal storage and provably computes a low-rank approximation of a solution. In particular, when all solutions have low rank, the algorithm converges to a solution. This algorithm, SketchyCGM, modifies a standard convex optimization scheme, the conditional gradient method, to store only a small randomized sketch of the matrix variable. After the optimization terminates, the algorithm extracts a low-rank approximation of the solution from the sketch. In contrast to nonconvex heuristics, the guarantees for SketchyCGM do not rely on statistical models for the problem data. Numerical work demonstrates the benefits of SketchyCGM over heuristics.
BibTeX
@InProceedings{pmlr-v54-yurtsever17a,
title = {{Sketchy Decisions: Convex Low-Rank Matrix Optimization with Optimal Storage}},
author = {Yurtsever, Alp and Udell, Madeleine and Tropp, Joel and Cevher, Volkan},
booktitle = {Proceedings of the 20th International Conference on Artificial Intelligence and Statistics},
pages = {1188--1196},
year = {2017},
editor = {Singh, Aarti and Zhu, Jerry},
volume = {54},
series = {Proceedings of Machine Learning Research},
month = {20--22 Apr},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v54/yurtsever17a/yurtsever17a.pdf},
url = {https://proceedings.mlr.press/v54/yurtsever17a.html},
abstract = {This paper concerns a fundamental class of convex matrix optimization problems. It presents the first algorithm that uses optimal storage and provably computes a low-rank approximation of a solution. In particular, when all solutions have low rank, the algorithm converges to a solution. This algorithm, SketchyCGM, modifies a standard convex optimization scheme, the conditional gradient method, to store only a small randomized sketch of the matrix variable. After the optimization terminates, the algorithm extracts a low-rank approximation of the solution from the sketch. In contrast to nonconvex heuristics, the guarantees for SketchyCGM do not rely on statistical models for the problem data. Numerical work demonstrates the benefits of SketchyCGM over heuristics.}
}