Vladimir Koltchinskii

Vladimir Koltchinskii studies the mathematics of statistical learning. His joint work with Karim Lounici gives concentration inequalities and expectation bounds for sample covariance operators of centered Gaussian observations.^[ref]

Their bounds separate two questions: the typical size of the operator norm error, and its fluctuations around that typical size. In finite-dimensional Euclidean space, their effective dimension is comparable to $\mathrm{tr}(\Sigma )/\Vert \Sigma \Vert$. If most covariance eigenvalues are small, this quantity can be much smaller than the ambient dimension, explaining why statistical difficulty need not track the raw number of coordinates.^[ref]