Rate of convergence for geometric inference based on the empirical Christoffel function

TL;DR

This paper analyzes the convergence rate of support estimation using the empirical Christoffel function, providing finite sample bounds and parameter selection guidelines based on concentration inequalities.

Contribution

It offers a detailed finite sample convergence analysis for Christoffel function-based support estimators, including parameter tuning and rate comparisons.

Findings

Finite sample bounds comparable to existing methods

Parameter selection based on sample size

Concentration inequalities for empirical Christoffel function

Abstract

We consider the problem of estimating the support of a measure from a finite, independent, sample. The estimators which are considered are constructed based on the empirical Christoffel function. Such estimators have been proposed for the problem of set estimation with heuristic justifications. We carry out a detailed finite sample analysis, that allows us to select the threshold and degree parameters as a function of the sample size. We provide a convergence rate analysis of the resulting support estimation procedure. Our analysis establishes that we may obtain finite sample bounds which are comparable to existing rates for different set estimation procedures. Our results rely on concentration inequalities for the empirical Christoffel function and on estimates of the supremum of the Christoffel-Darboux kernel on sets with smooth boundaries, that can be considered of independent interest.