On the Absence of Spurious Local Minima in Nonlinear Low-Rank Matrix Recovery Problems

Yingjie Bi, Javad Lavaei
Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, PMLR 130:379-387, 2021.

Abstract

The restricted isometry property (RIP) is a well-known condition that guarantees the absence of spurious local minima in low-rank matrix recovery problems with linear measurements. In this paper, we introduce a novel property named bound difference property (BDP) to study low-rank matrix recovery problems with nonlinear measurements. Using RIP and BDP jointly, we propose a new criterion to certify the nonexistence of spurious local minima in the rank-1 case, and prove that it leads to a much stronger theoretical guarantee than the existing bounds on RIP.

Cite this Paper

BibTeX
@InProceedings{pmlr-v130-bi21a, title = { On the Absence of Spurious Local Minima in Nonlinear Low-Rank Matrix Recovery Problems }, author = {Bi, Yingjie and Lavaei, Javad}, booktitle = {Proceedings of The 24th International Conference on Artificial Intelligence and Statistics}, pages = {379--387}, year = {2021}, editor = {Banerjee, Arindam and Fukumizu, Kenji}, volume = {130}, series = {Proceedings of Machine Learning Research}, month = {13--15 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v130/bi21a/bi21a.pdf}, url = {https://proceedings.mlr.press/v130/bi21a.html}, abstract = { The restricted isometry property (RIP) is a well-known condition that guarantees the absence of spurious local minima in low-rank matrix recovery problems with linear measurements. In this paper, we introduce a novel property named bound difference property (BDP) to study low-rank matrix recovery problems with nonlinear measurements. Using RIP and BDP jointly, we propose a new criterion to certify the nonexistence of spurious local minima in the rank-1 case, and prove that it leads to a much stronger theoretical guarantee than the existing bounds on RIP. } }
Endnote
%0 Conference Paper %T On the Absence of Spurious Local Minima in Nonlinear Low-Rank Matrix Recovery Problems %A Yingjie Bi %A Javad Lavaei %B Proceedings of The 24th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2021 %E Arindam Banerjee %E Kenji Fukumizu %F pmlr-v130-bi21a %I PMLR %P 379--387 %U https://proceedings.mlr.press/v130/bi21a.html %V 130 %X The restricted isometry property (RIP) is a well-known condition that guarantees the absence of spurious local minima in low-rank matrix recovery problems with linear measurements. In this paper, we introduce a novel property named bound difference property (BDP) to study low-rank matrix recovery problems with nonlinear measurements. Using RIP and BDP jointly, we propose a new criterion to certify the nonexistence of spurious local minima in the rank-1 case, and prove that it leads to a much stronger theoretical guarantee than the existing bounds on RIP.
APA
Bi, Y. & Lavaei, J.. (2021). On the Absence of Spurious Local Minima in Nonlinear Low-Rank Matrix Recovery Problems . Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 130:379-387 Available from https://proceedings.mlr.press/v130/bi21a.html.

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