Abstract
A general framework for principal component analysis (PCA) in the presence of heteroskedastic noise is introduced. We propose an algorithm called HeteroPCA, which involves iteratively imputing the diagonal entries of the sample covariance matrix to remove estimation bias due to heteroskedasticity. This procedure is computationally efficient and provably optimal under the generalized spiked covariance model. A key technical step is a deterministic robust perturbation analysis on singular subspaces, which can be of independent interest. The effectiveness of the proposed algorithm is demonstrated in a suite of problems in high-dimensional statistics, including singular value decomposition (SVD) under heteroskedastic noise, Poisson PCA, and SVD for heteroskedastic and incomplete data.
Funding Statement
The research of Anru Zhang was supported in part by NSF CAREER award DMS-1944904, NSF Grant DMS-1811868, and NIH Grant R01-GM131399-01.
The research of Tony Cai was supported in part by NSF Grants DMS-1712735 and DMS-2015259 and NIH Grants R01-GM129781 and R01-GM123056.
The research of Yihong Wu was supported in part by the NSF Grant CCF-1527105, an NSF CAREER award CCF-1651588, and an Alfred Sloan fellowship.
Citation
Anru R. Zhang. T. Tony Cai. Yihong Wu. "Heteroskedastic PCA: Algorithm, optimality, and applications." Ann. Statist. 50 (1) 53 - 80, February 2022. https://doi.org/10.1214/21-AOS2074
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