2019/12/03 by Dan Garber, Garber, Dan · 3 citations
Computer Science · Engineering · Physics and Astronomy · #Advanced X-ray Imaging Techniques #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Medical Image Segmentation Techniques #Optimization and Control (math.OC) #Sparse and Compressive Sensing Techniques
paper · pdf · doi:10.48550/arxiv.1912.01467
openalex publication_date 2019/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider convex optimization problems which are widely used as convex relaxations for low-rank matrix recovery problems. In particular, in several important problems, such as phase retrieval and robust PCA, the underlying assumption in many cases is that the optimal solution is rank-one. In this paper we consider a simple and natural sufficient condition on the objective so that the optimal solution to these relaxations is indeed unique and rank-one. Mainly, we show that under this condition, the standard Frank-Wolfe method with line-search (i.e., without any tuning of parameters whatsoever), which only requires a single rank-one SVD computation per iteration, finds an ε-approximated solution in only O(log1/ε) iterations (as opposed to the previous best known bound of O(1/ε)), despite the fact that the objective is not strongly convex. We consider several variants of the basic method with improved complexities, as well as an extension motivated by robust PCA, and finally, an extension to nonsmooth problems.