Mass localization

TL;DR

This paper studies the problem of identifying the smallest sets in a class that contain a specified proportion of a measure's mass, analyzing the convergence of empirical solutions to the true sets in a measured metric space.

Contribution

It introduces a formal framework for small sets and their size, and analyzes the convergence of empirical mass localization sets to the true sets.

Findings

Convergence of empirical sets to true mass localization sets.

Formal definition of small sets and their size.

Analysis of the convergence behavior in measured metric spaces.

Abstract

For a given class $\mathcal{F}$ of closed sets of a measured metric space $(E,d,\mu)$, we want to find the smallest element $B$ of the class $\mathcal{F}$ such that $\mu(B)\geq 1-\alpha$, for a given $0<\alpha<1$. This set $B$ \textit{localizes the mass} of $\mu$. Replacing the measure $\mu$ by the empirical measure $\mu_n$ gives an empirical smallest set $B_n$. The article introduces a formal definition of small sets (and their size) and study the convergence of the sets $B_n$ to $B$ and of their size.