On the non-asymptotic concentration of heteroskedastic Wishart-type matrix

This paper focuses on the non-asymptotic concentration of the heteroskedastic Wishart-type matrices. Suppose Z is a p(1)-by-p(2) random matrix and Z(ij) similar to N(0, sigma(2)(ij)) independently, we prove the expected spectral norm of Wishart matrix deviations (i.e., E parallel to ZZ(T) - EZZ(T)parallel to) is upper bounded by (1 + epsilon) {2 sigma(C sigma R) + sigma(2)(C) + C sigma(R sigma)* root log(p(1) boolean AND p(2)) +C-sigma*(2) log(p(1) boolean AND p(2))}, where sigma(2)(C) := max(j) Sigma(p1)(i=1) sigma(2)(ij), sigma(2)(R) := max(i) Sigma(p2)(j=1) sigma(2)(ij) and sigma(2)(*) := max(i,j) sigma(2)(ij). 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., sigma(ij) approximate to sigma(i) or sigma(ij) approximate to sigma(j)) 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.