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2022 On the non-asymptotic concentration of heteroskedastic Wishart-type matrix
T. Tony Cai, Rungang Han, Anru R. Zhang
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Electron. J. Probab. 27: 1-40 (2022). DOI: 10.1214/22-EJP758


This paper focuses on the non-asymptotic concentration of the heteroskedastic Wishart-type matrices. Suppose Z is a p1-by-p2random matrix and ZijN(0,σij2) independently, we prove the expected spectral norm of Wishart matrix deviations (i.e., EZZEZZ) is upper bounded by


where σC2:=maxji=1p1σij2, σR2:=maxij=1p2σij2 and σ2:=maxi,jσij2. A minimax lower bound is developed that matches this upper bound. Then, we derive the concentration inequalities, moments, and tail bounds for the heteroskedastic Wishart-type matrix under more general distributions, such as sub-Gaussian and heavy-tailed distributions. Next, we consider the cases where Z has homoskedastic columns or rows (i.e., σijσi or σijσj) and derive the rate-optimal Wishart-type concentration bounds. Finally, we apply the developed tools to identify the sharp signal-to-noise ratio threshold for consistent clustering in the heteroskedastic clustering problem.

Funding Statement

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 Rungang Han and Anru R. Zhang was supported in part by NSF CAREER-1944904, NSF DMS-1811868, and NIH R01-GM131399.


Download Citation

T. Tony Cai. Rungang Han. Anru R. Zhang. "On the non-asymptotic concentration of heteroskedastic Wishart-type matrix." Electron. J. Probab. 27 1 - 40, 2022.


Received: 4 January 2021; Accepted: 16 February 2022; Published: 2022
First available in Project Euclid: 25 February 2022

Digital Object Identifier: 10.1214/22-EJP758

Primary: 60B20
Secondary: 46B09

Keywords: concentration inequality , nonasymptotic bound , Random matrix , Wishart matrix


Vol.27 • 2022
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