VON NEUMANN ENTROPY PENALIZATION AND LOW-RANK MATRIX ESTIMATION

• 成果类型: Article
• 署名作者: Koltchinskii, Vladimir
• 署名单位: University System of Georgia; Georgia Institute of Technology
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/11-AOS926
• 发表日期: 2011
• 页码: 2936-2973
• 关键词: Empirical Processes; completion; INEQUALITY; RECOVERY

摘要

We study a problem of estimation of a Hermitian nonnegatively definite matrix rho of unit trace (e. g., a density matrix of a quantum system) based on n i.i.d. measurements (X-1, Y-1), ... , (X-n, Y-n), where Y-j = tr(rho X-j)+ xi(j), j = 1, ... , n, {X-j} being random i.i.d. Hermitian matrices and {xi(j)} being i.i.d. random variables with E(xi(j) vertical bar X-j) = 0. The estimator (rho) over cap (epsilon) := S is an element of S-argmin [n(-1) Sigma(n)(j=1) (Y-j - tr(SXj))(2) + epsilon tr(S log S)] is considered, where S is the set of all nonnegatively definite Hermitian m x m matrices of trace 1. The goal is to derive oracle inequalities showing how the estimation error depends on the accuracy of approximation of the unknown state rho by low-rank matrices.