Stratifying High Dimensional Data Based on Proximity to the Convex Hull Boundary

TL;DR

This paper introduces a quadratic programming method to stratify high-dimensional data points based on their proximity to the convex hull boundary, aiding in identifying extremal and boundary-adjacent points even in noisy datasets.

Contribution

The paper presents a novel quadratic program that encodes geometric relationships to efficiently determine boundary proximity in high-dimensional data.

Findings

The method effectively identifies points near the convex hull boundary.

Computational complexity grows linearly with data dimension.

Parallel computation enables handling large, high-dimensional datasets.

Abstract

The convex hull of a set of points, $C$, serves to expose extremal properties of $C$ and can help identify elements in $C$ of high interest. For many problems, particularly in the presence of noise, the true vertex set (and facets) may be difficult to determine. One solution is to expand the list of high interest candidates to points lying near the boundary of the convex hull. We propose a quadratic program for the purpose of stratifying points in a data cloud based on proximity to the boundary of the convex hull. For each data point, a quadratic program is solved to determine an associated weight vector. We show that the weight vector encodes geometric information concerning the point's relationship to the boundary of the convex hull. The computation of the weight vectors can be carried out in parallel, and for a fixed number of points and fixed neighborhood size, the overall computational complexity of the algorithm grows linearly with dimension. As a consequence, meaningful computations can be completed on reasonably large, high dimensional data sets.