# Probabilistic smallest enclosing ball in high dimensions via subgradient sampling

## TL;DR

This paper introduces a new algorithm for the probabilistic smallest enclosing ball problem in high-dimensional spaces, reducing computational complexity and enabling applications in kernel-based shape fitting methods.

## Contribution

It presents a novel combination of sampling and stochastic subgradient descent to improve the efficiency of pSEB algorithms in high dimensions.

## Key findings

- Reduces exponential dependence on dimension to linear in the algorithm.
- Extends support vector data description (SVDD) to probabilistic cases.
- Applicable to shape fitting in high-dimensional Hilbert spaces.

## Abstract

We study a variant of the median problem for a collection of point sets in high dimensions. This generalizes the geometric median as well as the (probabilistic) smallest enclosing ball (pSEB) problems. Our main objective and motivation is to improve the previously best algorithm for the pSEB problem by reducing its exponential dependence on the dimension to linear. This is achieved via a novel combination of sampling techniques for clustering problems in metric spaces with the framework of stochastic subgradient descent. As a result, the algorithm becomes applicable to shape fitting problems in Hilbert spaces of unbounded dimension via kernel functions. We present an exemplary application by extending the support vector data description (SVDD) shape fitting method to the probabilistic case. This is done by simulating the pSEB algorithm implicitly in the feature space induced by the kernel function.