Probabilistic tight frames and representation of Positive Operator-Valued Measures

摘要

In this paper we solve an open problem posed in [10] related to the representation of Positive Operator-Valued Measures by means of tight probabilistic frames in R-d. Also, we investigate how far is the closest probabilistic tight frame from a given probability measure where the distance used is the quadratic Wasserstein metric W-2 used for optimal transportation problem for measures. In particular, we study the optimization problem I(mu, K) := inf(nu epsilon T(K)) W-2(2)(mu, nu) v), where T(K) is the set of all probabilistic tight frames whose supports are contained in K = R-d or K = Sd-1. This problem is solved and its optimum is given when the mean vector of mu is zero. In the other cases, we give concise upper and lower bounds for I(mu, K). (C) 2018 Elsevier Inc. All rights reserved.