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Set-valued recursions arising from vantage-point trees

Congzao Dong, Alexander Marynych, Ilya Molchanov

TL;DR

This work analyzes set-valued recursions arising from vantage-point (vp) trees built from i.i.d. uniform points in a fixed convex body $K$ under an arbitrary norm. A set-valued Markov chain for the left boundary is shown to be Harris recurrent and to converge to a random ball polyhedron $X_\infty$, formed by a finite intersection of scaled balls, independent of $K$, which also yields a limit theorem for the leftmost path length $L_n$. The authors derive a detailed limit theory for the left boundary via a shift-and-scale construction, establish a stationary distribution $\pi_\infty$ for the boundary sets, and connect these results to the asymptotics of the leftmost path height, including explicit expressions involving random exponential sums. The findings provide new insights into the asymptotic geometry of random vp trees and the complexity of traversing their leftmost paths.

Abstract

We study vantage-point trees constructed using an independent sample from the uniform distribution on a fixed convex body $K$ in $(\mathbb{R}^d,\|\cdot\|)$, where $\|\cdot\|$ is an arbitrary norm on $\mathbb{R}^d$. We prove that a sequence of sets, associated with the left boundary of a vantage-point tree, forms a recurrent Harris chain on the space of convex bodies in $(\mathbb{R}^d,\|\cdot\|)$. The limiting object is a ball polyhedron, that is, an a.s.~finite intersection of closed balls in $(\mathbb{R}^d,\|\cdot\|)$ of possibly different radii. As a consequence, we derive a limit theorem for the length of the leftmost path of a vantage-point tree.

Set-valued recursions arising from vantage-point trees

TL;DR

This work analyzes set-valued recursions arising from vantage-point (vp) trees built from i.i.d. uniform points in a fixed convex body under an arbitrary norm. A set-valued Markov chain for the left boundary is shown to be Harris recurrent and to converge to a random ball polyhedron , formed by a finite intersection of scaled balls, independent of , which also yields a limit theorem for the leftmost path length . The authors derive a detailed limit theory for the left boundary via a shift-and-scale construction, establish a stationary distribution for the boundary sets, and connect these results to the asymptotics of the leftmost path height, including explicit expressions involving random exponential sums. The findings provide new insights into the asymptotic geometry of random vp trees and the complexity of traversing their leftmost paths.

Abstract

We study vantage-point trees constructed using an independent sample from the uniform distribution on a fixed convex body in , where is an arbitrary norm on . We prove that a sequence of sets, associated with the left boundary of a vantage-point tree, forms a recurrent Harris chain on the space of convex bodies in . The limiting object is a ball polyhedron, that is, an a.s.~finite intersection of closed balls in of possibly different radii. As a consequence, we derive a limit theorem for the length of the leftmost path of a vantage-point tree.
Paper Structure (4 sections, 10 theorems, 71 equations, 1 figure)

This paper contains 4 sections, 10 theorems, 71 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Convergence of random sets associated with vp trees
  3. Proof of Theorem \ref{['thm:main1']}
  4. The length of the leftmost path

Key Result

Proposition 2.1

Let $(I_h,x_{l_h})_{h\in\mathbb{N}_0}$, $I_0=K$, be a sequence of sets eq:I_h_def constructed from a random vp tree. Then the sequences $(I_h,x_{l_h})_{h\in\mathbb{N}_0}$ and $(J_h,y_h)_{h\in\mathbb{N}_0}$ have the same distribution. For every $h\in\mathbb{N}_0$, $y_{h}=\mathcal{U}_h(J_{h})$, where

Figures (1)

  • Figure 1: First twenty values of the chain $(X_{h})_{h\in\mathbb{N}_0}$ in $\mathbb{R}^2$ with the Euclidean norm and $\tau=4/7$. The chain starts at $B_1$ and returns to this state on Step 15. Each value of the chain is a finite intersection of translated and scaled unit balls.

Theorems & Definitions (20)

  • Proposition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Lemma 3.1
  • proof : Proof of Lemma \ref{['lem:harris']}
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 10 more