A minimization problem for probabilistic frames

In this paper we continue our work in [13] where the minimization problem I(μ):=infν∈T⁡W22(μ,ν),T being the set of probabilistic tight frames in Rd and W2 the quadratic Wasserstein metric for measures, was solved in the particular case when the mean vector of the probabilistic frame μ is zero. The p...

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Bibliographic Details
Published in:Applied and computational harmonic analysis 2020-09, Vol.49 (2), p.558-572
Main Authors: Loukili, S., Maslouhi, M.
Format: Article
Language:English
Subjects:
Frames
Optimal transport
Parseval frames
Probabilistic frames
Tight probabilistic frames
Wasserstein metric
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Summary:In this paper we continue our work in [13] where the minimization problem I(μ):=infν∈T⁡W22(μ,ν),T being the set of probabilistic tight frames in Rd and W2 the quadratic Wasserstein metric for measures, was solved in the particular case when the mean vector of the probabilistic frame μ is zero. The present work solves this problem in the general case, and in addition, establishes the uniqueness of the optimum and gives its explicit expression. The resolution of this problem is obtained by solving first the minimization problem Ip(μ):=infν∈Tp⁡W22(μ,ν), where Tp stands for the set of all Parseval probabilistic tight frames in Rd.
ISSN:1063-5203
1096-603X
DOI:10.1016/j.acha.2020.05.005