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On the notion of polynomial reach: a statistical application

Alejandro Cholaquidis, Antonio Cuevas, Leonardo Moreno

TL;DR

This work addresses the problem of recovering geometric functionals from inside samples by exploiting the polynomial-volume property of parallel sets: when $V(r)$ is a polynomial of degree at most $d$ on an interval $[0,{\mathbf R}]$, its coefficients encode $\mu(S)$ and $L(S)$. The authors propose two statistical strategies to handle the unknown polynomial reach ${\mathbf R}$: (i) a consistent estimator $\tilde{\mathbf R}$ derived from the $L^2$-projection error $G_n$ and a vanishing threshold, and (ii) a practical, infra-estimating grid-based algorithm that yields a conservative lower bound $\widehat{\mathbf R}$ with provable asymptotic behavior under mild regularity assumptions. They establish convergence rates for the estimated polynomial coefficients and validate the methods through numerical experiments on planar sets, highlighting that volume and boundary measures can be estimated from inner samples without smoothing parameters, albeit with greater error for higher-order terms and more complex geometries. The practical impact lies in enabling robust estimation of volume and boundary measures from inside data alone, with applications to shape inference and geometric data analysis where outside sampling or smoothing are undesirable or infeasible. The results connect to Federer's reach and Minkowski content, offering a flexible, data-driven approach to geometric quantification in higher dimensions.

Abstract

The volume function V(t) of a compact set S\in R^d is just the Lebesgue measure of the set of points within a distance to S not larger than t. According to some classical results in geometric measure theory, the volume function turns out to be a polynomial, at least in a finite interval, under a quite intuitive, easy to interpret, sufficient condition (called ``positive reach'') which can be seen as an extension of the notion of convexity. However, many other simple sets, not fulfilling the positive reach condition, have also a polynomial volume function. To our knowledge, there is no general, simple geometric description of such sets. Still, the polynomial character of $V(t)$ has some relevant consequences since the polynomial coefficients carry some useful geometric information. In particular, the constant term is the volume of S and the first order coefficient is the boundary measure (in Minkowski's sense). This paper is focused on sets whose volume function is polynomial on some interval starting at zero, whose length (that we call ``polynomial reach'') might be unknown. Our main goal is to approximate such polynomial reach by statistical means, using only a large enough random sample of points inside S. The practical motivation is simple: when the value of the polynomial reach , or rather a lower bound for it, is approximately known, the polynomial coefficients can be estimated from the sample points by using standard methods in polynomial approximation. As a result, we get a quite general method to estimate the volume and boundary measure of the set, relying only on an inner sample of points and not requiring the use any smoothing parameter. This paper explores the theoretical and practical aspects of this idea.

On the notion of polynomial reach: a statistical application

TL;DR

This work addresses the problem of recovering geometric functionals from inside samples by exploiting the polynomial-volume property of parallel sets: when is a polynomial of degree at most on an interval , its coefficients encode and . The authors propose two statistical strategies to handle the unknown polynomial reach : (i) a consistent estimator derived from the -projection error and a vanishing threshold, and (ii) a practical, infra-estimating grid-based algorithm that yields a conservative lower bound with provable asymptotic behavior under mild regularity assumptions. They establish convergence rates for the estimated polynomial coefficients and validate the methods through numerical experiments on planar sets, highlighting that volume and boundary measures can be estimated from inner samples without smoothing parameters, albeit with greater error for higher-order terms and more complex geometries. The practical impact lies in enabling robust estimation of volume and boundary measures from inside data alone, with applications to shape inference and geometric data analysis where outside sampling or smoothing are undesirable or infeasible. The results connect to Federer's reach and Minkowski content, offering a flexible, data-driven approach to geometric quantification in higher dimensions.

Abstract

The volume function V(t) of a compact set S\in R^d is just the Lebesgue measure of the set of points within a distance to S not larger than t. According to some classical results in geometric measure theory, the volume function turns out to be a polynomial, at least in a finite interval, under a quite intuitive, easy to interpret, sufficient condition (called ``positive reach'') which can be seen as an extension of the notion of convexity. However, many other simple sets, not fulfilling the positive reach condition, have also a polynomial volume function. To our knowledge, there is no general, simple geometric description of such sets. Still, the polynomial character of has some relevant consequences since the polynomial coefficients carry some useful geometric information. In particular, the constant term is the volume of S and the first order coefficient is the boundary measure (in Minkowski's sense). This paper is focused on sets whose volume function is polynomial on some interval starting at zero, whose length (that we call ``polynomial reach'') might be unknown. Our main goal is to approximate such polynomial reach by statistical means, using only a large enough random sample of points inside S. The practical motivation is simple: when the value of the polynomial reach , or rather a lower bound for it, is approximately known, the polynomial coefficients can be estimated from the sample points by using standard methods in polynomial approximation. As a result, we get a quite general method to estimate the volume and boundary measure of the set, relying only on an inner sample of points and not requiring the use any smoothing parameter. This paper explores the theoretical and practical aspects of this idea.
Paper Structure (13 sections, 8 theorems, 51 equations, 2 figures, 10 tables)

This paper contains 13 sections, 8 theorems, 51 equations, 2 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Some perspective and motivation. The notion of polynomial reach
  3. Consistent estimation of the polynomial reach
  4. A consistent estimator
  5. An algorithm for a lower bound of the polynomial reach
  6. On the assumptions H1 and H2 of Theorem \ref{['th:consist']}
  7. Estimation of the polynomial coefficients: convergence rates
  8. Computational aspects. Numerical experiments
  9. The case where $\mathbf{R}$ is small
  10. Estimation of the coefficients
  11. Some conclusions
  12. Proofs of Lemmas 1 and 2
  13. Tables

Key Result

Lemma 1

Let $S\subset\mathbb{R}^d$ be a compact set such that $V(0)>0$ and the right-hand side derivative $V^+(0)$ does exists and is finite. Let $\aleph_n=\{X_1,\ldots,X_n\}\subset S$ be a sequence of finite sets such that $\gamma_n=d_H(\aleph_n,S)\to 0$ and $2\gamma_n<b<+\infty$. Then, for all $n$ large e and As a consequence, given the compact interval $I=[0,b]$ we have that, for all $n$ large enough,

Figures (2)

  • Figure 1: In blue solid line, the boundary of the "Pacman set"; in red solid line the boundary of the corresponding dilation (parallel set) with radius $0.3$. The dilations, with radii $0.08$ and $0.3$, of a sample of size $n=1000$ correspond to the light blue region and the light blue+orange region, respectively.
  • Figure 2: In red solid line the boundary of the parallel set of radius $0.7$. In light blue it is shown the dilatation of a sample of size $n=3000$, of radius $0.1$, and the region in orange plus the region in light blue is the dilatation of the sample of radius 0.7 for the union of squares.

Theorems & Definitions (19)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Lemma 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • ...and 9 more