A varifold-type estimation for data sampled on a rectifiable set

TL;DR

This paper develops a kernel-based estimator for inferring varifold structures from i.i.d. samples on a possibly singular, piecewise smooth shape in Euclidean space, with proven convergence rates.

Contribution

It introduces a novel kernel-based method for estimating varifold structures from sampled data, accommodating non-uniform densities and singularities.

Findings

Estimator converges in expectation under bounded Lipschitz distance.

Convergence rate depends on shape dimension, regularity, and density smoothness.

Method handles shapes with singular sets of small measure.

Abstract

We investigate the inference of varifold structures in a statistical framework: assuming that we have access to i.i.d. samples in $\mathbb{R}^n$ obtained from an underlying $d$--dimensional shape $S$ endowed with a possibly non uniform density $\theta$, we propose and analyse an estimator of the varifold structure associated to $S$. The shape $S$ is assumed to be piecewise $C^{1,a}$ in a sense that allows for a singular set whose small enlargements are of small $d$--dimensional measure. The estimators are kernel--based both for infering the density and the tangent spaces and the convergence result holds for the bounded Lipschitz distance between varifolds, in expectation and in a noiseless model. The mean convergence rate involves the dimension $d$ of $S$, its regularity through $a \in (0, 1]$ and the regularity of the density $\theta$.