Almost sharp covariance and Wishart-type matrix estimation
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Let X_1,..., X_n ∈ℝ^d be independent Gaussian random vectors with independent entries and variance profile (b_ij)_i ∈ [d],j ∈ [n]. A major question in the study of covariance estimation is to give precise control on the deviation of ∑_j ∈ [n]X_jX_j^T-𝔼 X_jX_j^T. We show that under mild conditions, we have 𝔼∑_j ∈ [n]X_jX_j^T-𝔼 X_jX_j^T≲max_i ∈ [d](∑_j ∈ [n]∑_l ∈ [d]b_ij^2b_lj^2)^1/2+max_j ∈ [n]∑_i ∈ [d]b_ij^2+error. The error is quantifiable, and we often capture the 4th-moment dependency already presented in the literature for some examples. The proofs are based on the moment method and a careful analysis of the structure of the shapes that matter. We also provide examples showing improvement over the past works and matching lower bounds.
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