On the Non-Asymptotic Concentration of Heteroskedastic Wishart-type Matrix
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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∼ N(0,σ_ij^2) independently, we prove that ZZ^⊤ - ZZ^⊤≤ (1+ϵ){2σ_Cσ_R + σ_C^2 + Cσ_Rσ_*√(log(p_1 ∧ p_2)) + Cσ_*^2log(p_1 ∧ p_2)}, where σ_C^2 := max_j ∑_i=1^p_1σ_ij^2, σ_R^2 := max_i ∑_j=1^p_2σ_ij^2 and σ_*^2 := max_i,jσ_ij^2. 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., σ_ij≈σ_i or σ_ij≈σ_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.
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