Work

Set-valued recursions arising from vantage-point trees

Year: 2023

Type: preprint

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.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2312.05651

Authors: Dong, Congzao, Marynych, Alexander, Molchanov, Ilya

Institutions: Xidian University, Queen Mary University of London, Taras Shevchenko National University of Kyiv, University of Bern

Language: English

Topic: Point processes and geometric inequalities

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: Queen Mary University of London, Leverhulme Trust

OpenAlex ID: W4389650437

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