Hou, Thomas Y. and Li, Zhenzhen and Zhang, Ziyun (2019) Analysis of Asymptotic Escape of Strict Saddle Sets in Manifold Optimization. . (Unpublished) https://resolver.caltech.edu/CaltechAUTHORS:20200122-133158689
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Abstract
In this paper, we provide some analysis on the asymptotic escape of strict saddles in manifold optimization using the projected gradient descent (PGD) algorithm. One of our main contributions is that we extend the current analysis to include non-isolated and possibly continuous saddle sets with complicated geometry. We prove that the PGD is able to escape strict critical submanifolds under certain conditions on the geometry and the distribution of the saddle point sets. We also show that the PGD may fail to escape strict saddles under weaker assumptions even if the saddle point set has zero measure and there is a uniform escape direction. We provide a counterexample to illustrate this important point. We apply this saddle analysis to the phase retrieval problem on the low-rank matrix manifold, prove that there are only a finite number of saddles, and they are strict saddles with high probability. We also show the potential application of our analysis for a broader range of manifold optimization problems.
Item Type: | Report or Paper (Discussion Paper) | ||||||||
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Additional Information: | The research was in part supported by NSF Grants DMS-1613861, DMS-1907977 and DMS-1912654. | ||||||||
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Subject Keywords: | Manifold optimization, projected gradient descent, strict saddles, stable manifold theorem, phase retrieval | ||||||||
Classification Code: | AMS subject classifications: 58D17, 37D10, 65F10, 90C26 | ||||||||
DOI: | 10.48550/arXiv.1911.12518 | ||||||||
Record Number: | CaltechAUTHORS:20200122-133158689 | ||||||||
Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20200122-133158689 | ||||||||
Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||
ID Code: | 100844 | ||||||||
Collection: | CaltechAUTHORS | ||||||||
Deposited By: | Tony Diaz | ||||||||
Deposited On: | 22 Jan 2020 22:02 | ||||||||
Last Modified: | 02 Jun 2023 01:02 |
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