A hierarchy of convex relaxations for the total variation distance

Jean-Bernard Lasserre

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arXiv · 2401.01086

A hierarchy of convex relaxations for the total variation distance

Jean-Bernard Lasserre

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Abstract

Given two measures $\mu$, $\nu$ on Rd that satisfy Carleman's condition, we provide a numerical scheme to approximate as closely as desired the total variation distance between $\mu$ and $\nu$. 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 $\mu$-$\nu$.

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Jean-Bernard Lasserre. 2024-01-02. A hierarchy of convex relaxations for the total variation distance. https://arxiv.org/abs/2401.01086

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