Von Neumann Entropy Penalization and Low Rank Matrix Estimation

Understand

A problem of statistical estimation of a Hermitian nonnegatively definite matrix of unit trace (for instance, a density matrix in quantum state tomography) is studied.

• The approach is based on penalized least squares method with a complexity penalty defined in terms of von Neumann entropy.
• A number of oracle inequalities have been proved showing how the error of the estimator depends on the rank and other characteristics of the oracles.
• The methods of proofs are based on empirical processes theory and probabilistic inequalities for random matrices, in particular, noncommutative versions of Bernstein inequality.