A Sequential Computation Algorithm for the Center of the Smallest Enclosing Ball

TL;DR

This paper introduces a simple, sequential algorithm for computing the center of the smallest enclosing ball in high-dimensional space, leveraging the Arimoto-Blahut algorithm for efficient convergence.

Contribution

It develops a novel recurrence formula for barycentric coordinates of the SEB center, applicable even when an equidistant point does not exist, with proven convergence and practical efficiency.

Findings

Algorithm converges with complexity O(κ n^2 log(1/ε))

Performs well compared to conventional methods in speed and accuracy

Simple formula makes it practical for high-dimensional data

Abstract

In this paper, we consider the problem of finding the center $Q^\ast$ of the SEB (smallest enclosing ball) for $n$ points in $d$-dimensional Euclidean space. One application of the SEB is SVDD (support vector data description) in support vector machines. Our objective is to develop a sequential computation algorithm for determining the barycentric coordinate of $Q^\ast$. To achieve it, we apply the concept of the Arimoto-Blahut algorithm, which is a sequential computation algorithm used to compute the channel capacity. We first consider the case in which an equidistant point $\widetilde{Q}$ from the $n$ points exists, and construct a recurrence formula that converges to the barycentric coordinate $\widetilde{\bm\lambda}$ of $\widetilde{Q}$. When $\widetilde{Q}$ lies within the convex hull of the $n$ points, $\widetilde{Q}$ coincides with $Q^\ast$, hence in this case, the recurrence formula converges to the barycentric coordinate $\bm\lambda^\ast$ of $Q^\ast$. The resulting recurrence formula is very simple because it uses only the coordinates of the $n$ points. The computational complexity, with an approximation error of $\epsilon$ to the exact solution $\widetilde{\bm\lambda}$, is $O(\kappa n^2\log(1/\epsilon))$, where $\kappa$ is a value determined by the $n$ points. Furthermore, we modify the algorithm so that it can also be applied in cases where $\widetilde{Q}$ does not exist, and evaluate the convergence performance numerically. We compare the proposed algorithm with conventional algorithms in terms of run time and computational accuracy through several examples. The proposed algorithm has some advantages and some disadvantages compared to the conventional algorithms, but overall, since the proposed algorithm can be computed using a very simple formula, it is considered sufficiently practical.