Journey to the Center of the Point Set

TL;DR

This paper improves the algorithm for finding approximate centerpoints in high-dimensional data, reducing the complexity and simplifying the process, marking the first progress in over twenty years.

Contribution

An improved, simpler algorithm for computing a 1/d^2-centerpoint in $ ilde{O}(d^7)$ time, advancing the longstanding problem in high-dimensional computational geometry.

Findings

First progress in over twenty years on centerpoint computation.

New algorithm is simpler and faster, with $ ilde{O}(d^7)$ complexity.

Applicable to various problems in high-dimensional data analysis.

Abstract

$\renewcommand{\Re}{\mathbb{R}} \newcommand{\eps}{\varepsilon} \newcommand{\Net}{S} \newcommand{\tldO}{{\widetilde{O}}} \newcommand{\body}{C} $ We revisit an algorithm of Clarkson etal [CEMST96], that computes (roughly) a $1/(4d^2)$-centerpoint in $\tldO(d^9)$ time, for a point set in $\Re^d$, where $\tldO$ hides polylogarithmic terms. We present an improved algorithm that computes (roughly) a $1/d^2$-centerpoint with running time $\tldO(d^7)$. While the improvements are (arguably) mild, it is the first progress on this well known problem in over twenty years. The new algorithm is simpler, and the running time bound follows by a simple random walk argument, which we believe to be of independent interest. We also present several new applications of the improved centerpoint algorithm.