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A hierarchy of convex relaxations for the total variation distance

Jean-Bernard Lasserre

TL;DR

This work tackles the numerical evaluation of the total variation distance $\|\mu-\nu\|_{TV}$ between measures on $\mathbb{R}^d$ using only moment information. It recasts the problem as an infinite-dimensional linear program leveraging the Hahn–Jordan decomposition and imposes domination constraints to build a Generalized Moment Problem, which is then solved via a convergent Moment-SOS hierarchy of semidefinite relaxations. The hierarchy yields monotone, computable lower bounds $\rho_n$ that converge from below to $\|\mu-\nu\|_{TV}$, with the pseudo-moments converging to the Hahn–Jordan components; in the univariate discrete case, exact TV is recovered once the degree $n$ reaches $\max\{m_1,m_2\}$, where $m_1,m_2$ are the numbers of atoms. The approach works under Carleman’s condition (no compact support required) and remains applicable when moments are known in closed form or estimated from i.i.d. samples, providing a practical, mesh-free alternative to discretization-based methods for evaluating distributional distances.

Abstract

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 $μ$-$ν$.

A hierarchy of convex relaxations for the total variation distance

TL;DR

This work tackles the numerical evaluation of the total variation distance between measures on using only moment information. It recasts the problem as an infinite-dimensional linear program leveraging the Hahn–Jordan decomposition and imposes domination constraints to build a Generalized Moment Problem, which is then solved via a convergent Moment-SOS hierarchy of semidefinite relaxations. The hierarchy yields monotone, computable lower bounds that converge from below to , with the pseudo-moments converging to the Hahn–Jordan components; in the univariate discrete case, exact TV is recovered once the degree reaches , where are the numbers of atoms. The approach works under Carleman’s condition (no compact support required) and remains applicable when moments are known in closed form or estimated from i.i.d. samples, providing a practical, mesh-free alternative to discretization-based methods for evaluating distributional distances.

Abstract

Given two measures , on Rd that satisfy Carleman's condition, we provide a numerical scheme to approximate as closely as desired the total variation distance between and . 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 -.
Paper Structure (13 sections, 10 theorems, 55 equations, 2 tables)

This paper contains 13 sections, 10 theorems, 55 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Main result
  3. Notation and definitions
  4. A preliminary result
  5. Main result
  6. A convergent hierarchy of semidefinite relaxations
  7. A dual of \ref{['LP-relax']}
  8. Computational remarks
  9. Moment information
  10. Discrete (univariate) measures
  11. Numerical examples
  12. Conclusion
  13. Appendix

Key Result

Theorem 2.1

(CUP) Let a sequence $\boldsymbol{\mu}=(\mu_{{\boldsymbol{\alpha}}})_{{\boldsymbol{\alpha}}\in\mathbb{N}^d}$ be such that $\mathbf{M}_n(\boldsymbol{\mu})\succeq0$, for all $n\in\mathbb{N}$. If $\boldsymbol{\mu}$ satisfies Carleman's condition carleman then $\boldsymbol{\mu}$ has a representing measu

Theorems & Definitions (23)

  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Corollary 2.6
  • proof
  • Corollary 2.7
  • ...and 13 more