Computing the Approximate Convex Hull in High Dimensions

TL;DR

This paper introduces a high-dimensional convex hull approximation method with a time complexity independent of dimension, suitable for online applications requiring a balance between accuracy and simplicity.

Contribution

It presents a novel greedy algorithm for approximating convex hulls in high dimensions with a fixed number of vertices, achieving efficient computation.

Findings

Time complexity is independent of dimension

Method is effective for high-dimensional data

Suitable for online and real-time applications

Abstract

In this paper, an effective method with time complexity of $\mathcal{O}(K^{3/2}N^2\log \frac{K}{\epsilon_0})$ is introduced to find an approximation of the convex hull for $N$ points in dimension $n$, where $K$ is close to the number of vertices of the approximation. Since the time complexity is independent of dimension, this method is highly suitable for the data in high dimensions. Utilizing a greedy approach, the proposed method attempts to find the best approximate convex hull for a given number of vertices. The approximate convex hull can be a helpful substitute for the exact convex hull for on-line processes and applications that have a favorable trade off between accuracy and parsimony.