Session B4 - Learning Theory - Semi-plenary talk
July 15, 15:30 ~ 16:20 - Room B6
Estimation of Smooth Functionals of Covariance Operators
School of Mathematics, Georgia Institute of Technology, USA - email@example.com
We discuss a problem of estimation of a smooth functional of unknown covariance operator of a mean zero Gaussian random variable in a Hilbert space based on a sample of its i.i.d. observations in a setting when either the dimension of the space, or so called effective rank of the covariance operator are allowed to be large (although much smaller than the sample size). The main example is the problem of estimation of a linear functional of unknown eigenvector of the true covariance that corresponds to its largest eigenvalue (the top principal component). In this case, in a joint work with Matthias Loeffler and Richard Nickl, we proved a minimax lower bound on the risk of an arbitrary estimator of a linear functional, developed an estimator for which this bound is attained (which is not true for a standard estimator based on the top principal component of sample covariance due to its large bias) and also proved a Berry-Esseen type bound on the accuracy of approximation of its distribution by normal. This estimator is based on a further development of a bias reduction method initially introduced in our joint work with Karim Lounici. We will also discuss a more general approach to bias reduction and efficient estimation of smooth functionals of the unknown covariance (developed in our joint work with Mayya Zhilova).