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

Blanche Buet, Charly Boricaud

TL;DR

This work develops a kernel-based statistical framework to infer the varifold structure V_S associated with a possibly nonuniformly sampled, d-dimensional rectifiable set S in ℝ^n. It separates the problem into estimating the ambient d-dimensional measure 𝓗^d|_S via a density-corrected empirical measure and estimating tangent spaces through covariance-type operators, ultimately constructing varifold-type estimators that converge to W_S in the bounded Lipschitz distance. Under a piecewise Hölder regularity class that allows singular sets of controlled size, the paper derives explicit, uniform convergence rates for density, measure, tangent spaces, and the varifold itself, with rates of the form N^{−c/(d+2c)} where c = min(a,b). The results show minimax-type convergence for order-1 geometric information beyond smooth manifolds, and provide a pathway to estimating curvature-related quantities from noisy/inhomogeneous samples. These contributions have potential impacts on geometric inference, shape reconstruction, and curvature estimation from point clouds within low-regularity settings.

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 $θ$, 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 $θ$.

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

TL;DR

This work develops a kernel-based statistical framework to infer the varifold structure V_S associated with a possibly nonuniformly sampled, d-dimensional rectifiable set S in ℝ^n. It separates the problem into estimating the ambient d-dimensional measure 𝓗^d|_S via a density-corrected empirical measure and estimating tangent spaces through covariance-type operators, ultimately constructing varifold-type estimators that converge to W_S in the bounded Lipschitz distance. Under a piecewise Hölder regularity class that allows singular sets of controlled size, the paper derives explicit, uniform convergence rates for density, measure, tangent spaces, and the varifold itself, with rates of the form N^{−c/(d+2c)} where c = min(a,b). The results show minimax-type convergence for order-1 geometric information beyond smooth manifolds, and provide a pathway to estimating curvature-related quantities from noisy/inhomogeneous samples. These contributions have potential impacts on geometric inference, shape reconstruction, and curvature estimation from point clouds within low-regularity settings.

Abstract

We investigate the inference of varifold structures in a statistical framework: assuming that we have access to i.i.d. samples in obtained from an underlying --dimensional shape endowed with a possibly non uniform density , we propose and analyse an estimator of the varifold structure associated to . The shape is assumed to be piecewise in a sense that allows for a singular set whose small enlargements are of small --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 of , its regularity through and the regularity of the density .

Paper Structure

This paper contains 24 sections, 31 theorems, 296 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Reconstruction of the varifold structure
  3. Comments and perspectives
  4. Organisation of the paper
  5. Preliminary notions around rectifiability and varifolds
  6. Radon measures and Ahlfors regularity
  7. Rectifiability
  8. Varifolds
  9. Inference framework
  10. Some preliminary facts concerning the empirical measure
  11. Efron-Stein concentration inequality and pointwise estimator of density
  12. Estimation of the d--dimensional measure carried by S
  13. Nested partitions and mean discrepancy over partitions
  14. A technical mean convergence result for a Bounded Lipschitz type term
  15. Estimation of the measure carried by S
  16. ...and 9 more sections

Key Result

Proposition 2.8

Let $\mu$ be a Radon measure in $\mathbb{R}^n$, $S= \mathop{\mathrm{supp}}\limits (\mu)$. Assume that there exists $d >0, C_0\geq 1$ such that $\forall x \in S,$$\forall 0<r< {\rm diam}(S)$, Then, for all bounded set $B\subset \mathbb{R}^n$ and for all $\delta > 0$, the minimal number $m(B \cap S, \delta)$ of sets of diameter smaller than $\delta$ needed to form a partition of $B\cap S$ satisfies

Figures (2)

  • Figure 1:
  • Figure :

Theorems & Definitions (93)

  • Definition 2.1: Radon measure
  • Definition 2.2: Weak star convergence of Radon measures
  • Definition 2.3: Bounded Lipschitz distance
  • Definition 2.4: Localized Bounded Lipschitz distance
  • Definition 2.5: Ahlfors regularity
  • Remark 2.6: Finite Ahlfors regular measure
  • Example 2.7: $d$--Ahlfors measures.
  • Proposition 2.8
  • Remark 2.9
  • proof
  • ...and 83 more