Open Access
October 2021 Bridging convex and nonconvex optimization in robust PCA: Noise, outliers and missing data
Yuxin Chen, Jianqing Fan, Cong Ma, Yuling Yan
Author Affiliations +
Ann. Statist. 49(5): 2948-2971 (October 2021). DOI: 10.1214/21-AOS2066


This paper delivers improved theoretical guarantees for the convex programming approach in low-rank matrix estimation, in the presence of (1) random noise, (2) gross sparse outliers and (3) missing data. This problem, often dubbed as robust principal component analysis (robust PCA), finds applications in various domains. Despite the wide applicability of convex relaxation, the available statistical support (particularly the stability analysis in the presence of random noise) remains highly suboptimal, which we strengthen in this paper. When the unknown matrix is well conditioned, incoherent and of constant rank, we demonstrate that a principled convex program achieves near-optimal statistical accuracy, in terms of both the Euclidean loss and the loss. All of this happens even when nearly a constant fraction of observations are corrupted by outliers with arbitrary magnitudes. The key analysis idea lies in bridging the convex program in use and an auxiliary nonconvex optimization algorithm, and hence the title of this paper.

Funding Statement

Y. Chen is supported in part by the AFOSR YIP award FA9550-19-1-0030, by the ONR Grant N00014-19-1-2120, by the ARO Grants W911NF-20-1-0097 and W911NF-18-1-0303, by NSF Grants CCF-1907661, IIS-1900140, IIS-2100158, and DMS-2014279 and by the Princeton SEAS innovation award. J. Fan is supported in part by the NSF Grants DMS-1662139 and DMS-1712591, the ONR Grant N00014-19-1-2120 and the NIH Grant 2R01-GM072611-14.


Author names are sorted alphabetically. Y. Chen is the corresponding author.


Download Citation

Yuxin Chen. Jianqing Fan. Cong Ma. Yuling Yan. "Bridging convex and nonconvex optimization in robust PCA: Noise, outliers and missing data." Ann. Statist. 49 (5) 2948 - 2971, October 2021.


Received: 1 January 2020; Revised: 1 September 2020; Published: October 2021
First available in Project Euclid: 12 November 2021

MathSciNet: MR4338899
zbMATH: 1486.62181
Digital Object Identifier: 10.1214/21-AOS2066

Primary: 62F10
Secondary: 62B10

Keywords: convex relaxation , ℓ∞ guarantees , leave-one-out analysis , Robust principal component analysis

Rights: Copyright © 2021 Institute of Mathematical Statistics

Vol.49 • No. 5 • October 2021
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