大阪大学　数理・データ科学教育研究センター
Center for Mathematical Modeling and Data Science,Osaka University

Bootstrap confidence sets for spectral projectors of sample covariance

Vladimir Ulyanov (Moscow State University)

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大阪大学 数理・データ科学セミナー　データ科学セミナーシリーズ 第15回　(第79回データ科学特別セミナー 共催)

Bootstrap confidence sets for spectral projectors of sample covariance

Vladimir Ulyanov (Moscow State University)

Let X_1, ... ,X_n be i.i.d. sample in R^p with zero mean and the covariance matrix S. The problem of recovering the projector onto the eigenspace of S from these observations naturally arises in many applications. Recent technique from [Koltchinskii and Lounici, 2015b] helps to study the asymptotic distribution of the distance in the Frobenius norm between the true projector P_r on the subspace of the r th eigenvalue and its empirical counterpart \hat{P}_r in terms of the effective trace of S. This paper offers a bootstrap procedure for building sharp condence sets for the true projector P_r from the given data. This procedure does not rely on the asymptotic distribution of || P_r - \hat{P}_r ||_2 and its moments, it applies for small or moderate sample size n and large dimension p . The main result states the validity of the proposed procedure for nite samples with an explicit error bound on the error of bootstrap approximation. This bound involves some new sharp results on Gaussian comparison and Gaussian anti-concentration in high dimension. Numeric results confirm a nice performance of the method in realistic examples.

These are the joint results with Prof.V.Spokoiny (WIAS, Berlin, Germany) and Dr A.Naumov (Skoltech, Moscow, Russia)