This paper focuses on the non-asymptotic concentration of the heteroskedastic Wishart-type matrices. Suppose Z is a -by-random matrix and independently, we prove the expected spectral norm of Wishart matrix deviations (i.e., ) is upper bounded by
where , and . 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., or ) 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.
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.
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. https://doi.org/10.1214/22-EJP758