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On the expected number of facets for the convex hull of samples

Feng Zhao, Xinyi Tong, Shao-Lun Huang

Abstract

This paper studies the convex hull of $d$-dimensional samples i.i.d. generated from spherically symmetric distributions. Specifically, we derive a complete integration formula for the expected facet number of the convex hull. This formula is with respect to the CDF of the radial distribution. As the number of samples approaches infinity, the integration formula enables us to obtain the asymptotic value of the expected facet number for three categories of spherically symmetric distributions. Additionally, the asymptotic result can be applied to estimating the sample complexity in order that the probability measure of the convex hull tends to one.

On the expected number of facets for the convex hull of samples

Abstract

This paper studies the convex hull of -dimensional samples i.i.d. generated from spherically symmetric distributions. Specifically, we derive a complete integration formula for the expected facet number of the convex hull. This formula is with respect to the CDF of the radial distribution. As the number of samples approaches infinity, the integration formula enables us to obtain the asymptotic value of the expected facet number for three categories of spherically symmetric distributions. Additionally, the asymptotic result can be applied to estimating the sample complexity in order that the probability measure of the convex hull tends to one.
Paper Structure (13 sections, 11 theorems, 41 equations, 1 figure, 1 table)

This paper contains 13 sections, 11 theorems, 41 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Related work
  3. Integration formula
  4. Three distribution families
  5. Distributions with polynomial tails
  6. Distributions with exponential tails
  7. Distributions with truncated tails
  8. Sample complexity in data interpolation
  9. Proof of Theorem \ref{['thm:H']}
  10. Computations involved in three distribution tails
  11. Computations involved in polynomial tails
  12. Computations involved in exponential tails
  13. Computations involved in truncated tails

Key Result

Theorem 1

For $d\geq 2$, the integration formula for $H(x)$ is given as where the auxiliary function $K(x)$ is defined in the following way:

Figures (1)

  • Figure 1: The length of red line represents $a_1$ while the green line corresponds to $|a_2|$.

Theorems & Definitions (13)

  • Theorem 1
  • Definition 1
  • Theorem 2
  • Example 1
  • Theorem 3
  • Theorem 4
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • ...and 3 more