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.
Keywords: Support estimation, christoffel function, concentration, finite sample, convergence rate
@article{PS_2022__26_1_171_0,
author = {Vu, Mai Trang and Bachoc, Fran\c{c}ois and Pauwels, Edouard},
title = {Rate of convergence for geometric inference based on the empirical {Christoffel} function},
journal = {ESAIM: Probability and Statistics},
pages = {171--207},
year = {2022},
publisher = {EDP-Sciences},
volume = {26},
doi = {10.1051/ps/2022003},
mrnumber = {4387184},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ps/2022003/}
}
TY - JOUR AU - Vu, Mai Trang AU - Bachoc, François AU - Pauwels, Edouard TI - Rate of convergence for geometric inference based on the empirical Christoffel function JO - ESAIM: Probability and Statistics PY - 2022 SP - 171 EP - 207 VL - 26 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/ps/2022003/ DO - 10.1051/ps/2022003 LA - en ID - PS_2022__26_1_171_0 ER -
%0 Journal Article %A Vu, Mai Trang %A Bachoc, François %A Pauwels, Edouard %T Rate of convergence for geometric inference based on the empirical Christoffel function %J ESAIM: Probability and Statistics %D 2022 %P 171-207 %V 26 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/ps/2022003/ %R 10.1051/ps/2022003 %G en %F PS_2022__26_1_171_0
Vu, Mai Trang; Bachoc, François; Pauwels, Edouard. Rate of convergence for geometric inference based on the empirical Christoffel function. ESAIM: Probability and Statistics, Tome 26 (2022), pp. 171-207. doi: 10.1051/ps/2022003
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