A hierarchy of convex relaxations for the total variation distance
成果类型：
Article; Early Access
署名作者：
Lasserre, Jean B.
署名单位：
Centre National de la Recherche Scientifique (CNRS); Universite de Toulouse; Universite Toulouse 1 Capitole; Toulouse School of Economics
刊物名称：
MATHEMATICAL PROGRAMMING
ISSN/ISSBN：
0025-5610
DOI：
10.1007/s10107-025-02293-2
发表日期：
2025
关键词：
摘要：
Given two measures mu,nu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu ,\nu $$\end{document} on Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}<^>d$$\end{document} that satisfy Carleman's condition, we provide a numerical scheme to approximate as closely as desired the total variation distance between mu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} and nu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document}. (In particular, the supports of mu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} and nu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} are not necessarily compact.) 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. 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu -\nu $$\end{document}. Illustrative examples are provided.
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