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Conditions de Monge, Transport Optimal et Pont Relationnel : propriétés, applications et extension du couplage d'indétermination

Abstract : The starting point of this thesis is the justification of the restriction to two canonical divergences in a problem of projection of a probability law towards a space with constrained margins. The first leads to the coupling of independence, the second to the so-called indeterminacy. The object of the thesis is the study of the second coupling. The indeterminacy coupling is first seen as an equilibrium thanks to its so-called relational coding. By rewriting the Monge property that it verifies, a decomposition of a draw is proposed and leads to a property of couple matching minimization between two successive realizations. It is applied to two problems: that of the spy and the partitioning of tasks. In the graph clustering problem, the usual modularity is seen as a deviation from independence and a modularity of indeterminacy is coined. The similarities and differences between both are studied on Gilbert's graphs. A review of the correlation criteria shows that they are written as a deviation from one or the other of the canonical equilibria. A general form emerges and reveals a common dot product which encodes the correlation. A theoretical distribution of this dot product is established.The indeterminacy is extended in the continuous domain and so is the notion of couple matching which is transposed into the so-called mean likelihood. It is shown that an associated indetermination copula can only be defined locally. Eventually, a statistical test to distinguish between the two equilibria is constructed and analyzed.
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Submitted on : Wednesday, March 9, 2022 - 2:16:08 PM
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  • HAL Id : tel-03602842, version 1

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Pierre Bertrand. Conditions de Monge, Transport Optimal et Pont Relationnel : propriétés, applications et extension du couplage d'indétermination. Statistiques [math.ST]. Sorbonne Université, 2021. Français. ⟨NNT : 2021SORUS368⟩. ⟨tel-03602842⟩

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