Statistical inference for large matrices

Abstract

This thesis covers two topics on matrix analysis and estimation in machine learning and statistics. The first topic is about density matrix estimation with application in quantum state tomography. The density matrices are positively semi-definite Hermitian matrices of unit trace that describe the state of a quantum system. We develop minimax lower bounds on error rates of estimation of low rank density matrices in trace regression models used in quantum state tomography (in particular, in the case of Pauli measurements) with explicit dependence of the bounds on the rank and other complexity parameters. Such bounds are established for several statistically relevant distances, including quantum versions of Kullback-Leibler divergence (relative entropy distance) and of Hellinger distance (so called Bures distance), and Schatten p-norm distances. Sharp upper bounds and oracle inequalities for least squares estimator with von Neumann entropy penalization are obtained showing that minimax lower bounds are attained (up to logarithmic factors) for these distances. Another topic is about the analysis of the spectral perturbations of matrices under Gaussian noise. Given a matrix contaminated with Gaussian noise, we develop sharp upper bounds on the perturbation of linear forms of singular vectors. In particular, sharp upper bounds are proved for the component-wise perturbation of singular vectors. These results can be applied on sub-matrices localization and spectral clustering algorithms.