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Vladimir V. Ulyanov教授学术报告:Bootstrap confidence sets for spectral projectors of sample covariance

报告题目: Bootstrap confidence sets for spectral projectors of samplecovariance

报 告 人: Vladimir V. Ulyanov(莫斯科大学)

报告时间:2018102416:00-17:00

报告地点:知新楼B-1238

 

报告摘要:

Let X1, .. . , Xn be i.i.d. sample in Rp with zero mean and the covariance matrix\Sigma. The problem of recovering the projector onto an eigenspace of \Sigmafrom these observations naturally arises in many applications. Recent techniquedeveloped by Koltchinskii and Lounici helps to study the asymptoticdistribution of the distance in the Frobenius norm ||Pr – \hat{Pr}||_2 betweenthe true projector Pr on the subspace of the r-th eigenvalue and its empiricalcounterpart Pr in terms of the effective rank of \Sigma. In the talk we offer abootstrap procedure for building sharp confidence sets for the true projectorPr from the given data. This procedure does not rely on the asymptoticdistribution of ||Pr – \hat{Pr}||_2and its moments. It could be applied forsmall or moderate sample size n and large dimension p. The main result statesthe validity of the proposed procedure for Gaussian samples with an expliciterror bound for the error of bootstrap approximation. This bound involves somenew sharp results on Gaussian comparison and Gaussian anti-concentration inhigh-dimensional spaces. Numeric results confirm a good performance of the methodin realistic examples.

These arethe joint results with V.Spokoiny (WIAS, Berlin) and A.Naumov (HSE, Moscow).See details in

 

A.Naumov, V. Spokoiny, and V. Ulyanov, Bootstrap confidence sets for spectralprojectors of sample covariance (2017). arXiv:1703.00871.

 

 

欢迎各位老师同学积极参加!