2020/08/28 by Tommaso Cai, T. Tony Cai, Cai, T. Tony +4 · 1 citation
Mathematics · #Combinatorics #Eigenvalues and eigenvectors #FOS: Mathematics #Mathematical Inequalities and Applications #Mathematical analysis #Mathematics #Matrix (chemical analysis) #Physics #Point processes and geometric inequalities #Probability (math.PR) #Quantum mechanics #Random Matrices and Applications #Random matrix #Sigma #Statistics #Statistics Theory (math.ST) #Upper and lower bounds #Wishart distribution #math.PR #math.ST #stat.TH
paper · pdf · doi:10.48550/arxiv.2008.12434
Electronic Journal of Probability, to appear
openalex publication_date 2020/08/28 · arxiv created 2022/02/16 · arxiv updated 2022/02/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
This paper focuses on the non-asymptotic concentration of the heteroskedastic Wishart-type matrices. Suppose Z is a p1-by-p2 random matrix and Zij ∼ N(0,σij2) independently, we prove the expected spectral norm of Wishart matrix deviations (i.e., 𝔼 ‖ZZ^\top - 𝔼 ZZ^\top‖) is upper bounded by \beginsplit (1+ε)\2σCσR + σC2 + CσRσ_*√(log(p1 \wedge p2)) + Cσ_*2log(p1 \wedge p2)\, \endsplit where σC2 := maxj ∑i=1p1σij2, σR2 := maxi ∑j=1p2σij2 and σ_*2 := 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., σij ≈ σi or σij ≈ σj) 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.