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Computing Enclosing Depth

Bernd Gärtner, Fatime Rasiti, Patrick Schnider

TL;DR

This work defines enclosing depth $ED(S,q)$ as a robust combinatorial depth measure and provides the first algorithms to compute it in any dimension. The planar result achieves $O(n\log n)$ time via radial ordering and interval-based tests, while higher dimensions are handled by a halfspace decomposition approach with runtime $O(n^{d^2})$. The contributions establish enclosing depth as a computable lower bound for other depth notions and connect it to geometric primitives like halfspaces and transversals. The findings have implications for robust multivariate statistics and geometric data analysis, with open questions on closing the gap to the $O(n^{d-1})$ benchmarks seen for related depth measures and extending efficiency to broader problems.

Abstract

Enclosing depth is a recently introduced depth measure which gives a lower bound to many depth measures studied in the literature. So far, enclosing depth has only been studied from a combinatorial perspective. In this work, we give the first algorithms to compute the enclosing depth of a query point with respect to a data point set in any dimension. In the plane we are able to optimize the algorithm to get a runtime of O(n log n). In constant dimension, our algorithms still run in polynomial time.

Computing Enclosing Depth

TL;DR

This work defines enclosing depth as a robust combinatorial depth measure and provides the first algorithms to compute it in any dimension. The planar result achieves time via radial ordering and interval-based tests, while higher dimensions are handled by a halfspace decomposition approach with runtime . The contributions establish enclosing depth as a computable lower bound for other depth notions and connect it to geometric primitives like halfspaces and transversals. The findings have implications for robust multivariate statistics and geometric data analysis, with open questions on closing the gap to the benchmarks seen for related depth measures and extending efficiency to broader problems.

Abstract

Enclosing depth is a recently introduced depth measure which gives a lower bound to many depth measures studied in the literature. So far, enclosing depth has only been studied from a combinatorial perspective. In this work, we give the first algorithms to compute the enclosing depth of a query point with respect to a data point set in any dimension. In the plane we are able to optimize the algorithm to get a runtime of O(n log n). In constant dimension, our algorithms still run in polynomial time.
Paper Structure (4 sections, 12 theorems, 1 equation, 4 figures)

This paper contains 4 sections, 12 theorems, 1 equation, 4 figures.

Table of Contents

  1. Introduction
  2. The planar case
  3. Higher dimensions
  4. Conclusion

Key Result

Lemma 1

Let $S=\{s_1,\ldots,s_n\}\subset\mathbb{R}^2$ be a data point set and $q\in\mathbb{R}^2$ a query point such that $S\cup\{q\}$ is in general position. Denote by $S'=\{s'_1,\ldots,s'_n\}$ the point set defined by centrally projecting each point in $S$ to a circle of unit radius with center $q$. Then $

Figures (4)

  • Figure 1: The set $S = S_1 \cup S_2 \cup S_3$ 5-encloses the query point $q$ in $\mathbb{R}^2$.
  • Figure 2: The enclosing depth of the query point $q$ is at least 5.
  • Figure 3: The opposite neighbors $s^{(r)}$ and $s^{(\ell)}$ of a point $s\in S$.
  • Figure 4: An illustration of the proof of Lemma \ref{['fig:nbs_proof']}.

Theorems & Definitions (24)

  • Definition 1: Depth measure
  • Definition 2: k-enclosing
  • Definition 3: Enclosing Depth
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 4
  • ...and 14 more