Abstract

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 ( μ , K ) : = inf ν ∈ T ( K ) ⁡ W 2 2 ( μ , ν ) , where T ( K ) is the set of all probabilistic tight frames whose supports are contained in K = R d or K = S d − 1 . This problem is solved and its optimum is given when the mean vector of μ is zero. In the other cases, we give concise upper and lower bounds for I ( μ , K ) .

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