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arXiv · 2501.16315

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

Charly Boricaud·Blanche Buet
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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 $\theta$.

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Charly Boricaud, Blanche Buet. 2025-01-27. A varifold-type estimation for data sampled on a rectifiable set. https://arxiv.org/abs/2501.16315

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