Von Neumann entropy penalization and low-rank matrix estimation
Author:
- Vladimir KoltchinskiiVKVladimir Koltchinskii
Open Access
- 1 December 2011
- journal article
- Published by Institute of Mathematical Statistics in The Annals of Statistics
- Vol. 39 (6) , 2936-2973
Abstract
We study a problem of estimation of a Hermitian nonnegatively definite matrix ρ of unit trace (e.g., a density matrix of a quantum system) based on n i.i.d. measurements (X1, Y1), …, (Xn, Yn), where Yj = tr(ρXj) + ξj, j = 1, …, n, {Xj} being random i.i.d. Hermitian matrices and {ξj} being i.i.d. random variables with . The estimator is considered, where is the set of all nonnegatively definite Hermitian m × 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 ρ by low-rank matrices.Keywords
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