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Estimating the Convex Hull of the Image of a Set with Smooth Boundary: Error Bounds and Applications

Thomas Lew, Riccardo Bonalli, Lucas Janson, Marco Pavone

TL;DR

When applied to the problem of geometric inference from a random sample, the results give tighter and more general error bounds than the state of the art.

Abstract

We study the problem of estimating the convex hull of the image $f(X)\subset\mathbb{R}^n$ of a compact set $X\subset\mathbb{R}^m$ with smooth boundary through a smooth function $f:\mathbb{R}^m\to\mathbb{R}^n$. Assuming that $f$ is a submersion, we derive a new bound on the Hausdorff distance between the convex hull of $f(X)$ and the convex hull of the images $f(x_i)$ of $M$ sampled inputs $x_i$ on the boundary of $X$. When applied to the problem of geometric inference from a random sample, our results give error bounds that are tighter and more general than in previous work. We present applications to the problems of robust optimization, of reachability analysis of dynamical systems, and of robust trajectory optimization under bounded uncertainty.

Estimating the Convex Hull of the Image of a Set with Smooth Boundary: Error Bounds and Applications

TL;DR

When applied to the problem of geometric inference from a random sample, the results give tighter and more general error bounds than the state of the art.

Abstract

We study the problem of estimating the convex hull of the image of a compact set with smooth boundary through a smooth function . Assuming that is a submersion, we derive a new bound on the Hausdorff distance between the convex hull of and the convex hull of the images of sampled inputs on the boundary of . When applied to the problem of geometric inference from a random sample, our results give error bounds that are tighter and more general than in previous work. We present applications to the problems of robust optimization, of reachability analysis of dynamical systems, and of robust trajectory optimization under bounded uncertainty.
Paper Structure (40 sections, 32 theorems, 61 equations, 11 figures)

This paper contains 40 sections, 32 theorems, 61 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Notations and background
  3. Geometric regularity: reach, $R$-convexity, rolling balls, and $R$-smoothness
  4. Equivalences and connections between definitions
  5. Smoothness of the convex hull $\textrm{H}(\mathcal{Y})$
  6. Smoothness properties are preserved under diffeomorphisms
  7. Rolling-ball properties are preserved under submersions
  8. Smoothness properties are preserved when taking the convex hull
  9. Proof of Theorem \ref{['thm:submersion_rsmooth']} ($\textrm{H}(\mathcal{Y})$ is $R$-smooth)
  10. Error bounds
  11. Naive error bound via covering ($f$ is Lipschitz)
  12. Error bound via smoothness of the boundary ($f$ is $C^1$ and $\mathcal{X}$ is $r$-smooth)
  13. Bound on $d_{T_y\partial\textrm{H}(\mathcal{Y})}$ implies bound on Hausdorff distance (proof of Lemma \ref{['lem:tangent_implies_hausdorff']})
  14. Bound on $d_{T_y\partial\textrm{H}(\mathcal{Y})}$ if $f$ is a submersion (proof of Lemma \ref{['lem:bound_delta_Tphull']})
  15. Hausdorff distance error bound (proof of Theorem \ref{['thm:error_bound:smooth_boundary']})
  16. ...and 25 more sections

Key Result

Theorem 1.1

Let $r,\delta>0$, $\mathcal{X}\subset\mathbb{R}^m$ be a non-empty path-connected compact set that is $r$-smooth (see Definition def:rsmooth), $f:\mathbb{R}^m\to\mathbb{R}^n$, $\mathcal{Y}=f(\mathcal{X})$, and $Z_\delta=\{x_i\}_{i=1}^M\subset\partial\mathcal{X}$ be a $\delta$-cover of the boundary $\

Figures (11)

  • Figure 1: (a) Reach and medial axis. (b) A ball of radius $R$ rolls freely in $\mathcal{X}$ and in $\overline{\mathcal{X}^{\mathsf{c}}}$, so $\mathcal{X}$ is $R$-smooth.
  • Figure 2: The boundary of images of sets with smooth boundary may not be smooth due to self-intersections.
  • Figure 3: Positive reach is conserved under diffeomorphisms.
  • Figure 4: Definitions for the proof of Lemma \ref{['lem:submersion_rolling_ball_Y']}.
  • Figure 5: Decomposition of the boundary of $\textrm{H}(\mathcal{Y})$.
  • ...and 6 more figures

Theorems & Definitions (70)

  • Theorem 1.1: Estimation error for the convex hull of $f(\mathcal{X})$
  • Definition 2.1: Reach
  • Definition 2.2: $R$-convexity
  • Definition 2.3: Rolling ball
  • Definition 2.4: $R$-smooth set
  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • ...and 60 more