Von Neumann Entropy Penalization and Low Rank Matrix Estimation

TL;DR

This paper introduces a new penalized least squares method using von Neumann entropy for estimating positive semidefinite matrices, with theoretical error bounds depending on matrix rank and properties.

Contribution

It proposes a novel entropy-based penalty for low-rank matrix estimation and provides rigorous oracle inequalities for the estimator's error.

Findings

Derived oracle inequalities linking error to matrix rank and complexity

Utilized empirical process theory and noncommutative Bernstein inequalities

Established theoretical guarantees for the proposed estimation method

Abstract

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.