TY - JOUR
T1 - Bridging convex and nonconvex optimization in robust PCA
T2 - Noise, outliers and missing data
AU - Chen, Yuxin
AU - Fan, Jianqing
AU - Ma, Cong
AU - Yan, Yuling
N1 - Funding Information:
Funding. 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.
Publisher Copyright:
© Institute of Mathematical Statistics, 2021.
PY - 2021/10
Y1 - 2021/10
N2 - 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 ℓ∞ 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.
AB - 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 ℓ∞ 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.
KW - Convex relaxation
KW - Leave-one-out analysis
KW - Robust principal component analysis
KW - ℓ guarantees
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U2 - 10.1214/21-AOS2066
DO - 10.1214/21-AOS2066
M3 - Article
AN - SCOPUS:85117547746
SN - 0090-5364
VL - 49
SP - 2948
EP - 2971
JO - Annals of Statistics
JF - Annals of Statistics
IS - 5
ER -