Article Dans Une Revue Mathematical Programming, Series A Année : 2025

A hierarchy of convex relaxations for the total variation distance

Résumé

Given two measures µ, ν on Rd that satisfy Carleman's condition, we provide a numerical scheme to approximate as closely as desired the total variation distance between µ and ν. It consists of solving a sequence (hierarchy) of convex relaxations whose associated sequence of optimal values converges to the total variation distance, an additional illustration of the versatility of the Moment-SOS hierarchy. Indeed each relaxation in the hierarchy is a semidefinite program whose size increases with the number of involved moments. It has an optimal solution which is a couple of degree-2n pseudo-moments which converge, as n grows, to moments of the Hahn-Jordan decomposition of µ-ν.

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Dates et versions

hal-04367575 , version 1 (30-12-2023)
hal-04367575 , version 2 (02-05-2024)
hal-04367575 , version 3 (20-09-2024)
hal-04367575 , version 4 (15-10-2025)

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Jean-Bernard Lasserre. A hierarchy of convex relaxations for the total variation distance. Mathematical Programming, Series A, inPress, 8 p. ⟨10.1007/s10107-025-02293-2⟩. ⟨hal-04367575v4⟩
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