Shape Proportions and Sphericity in n Dimensions

TL;DR

This paper introduces two novel shape metrics, hyper-Sphericity and hyper-Shape Proportion, for quantifying the shape of objects in any number of dimensions, with applications to high-dimensional data analysis.

Contribution

The paper develops and explores two new shape metrics applicable to objects in arbitrary dimensions, extending previous dimension-specific metrics.

Findings

Hyper-Sphericity and hyper-Shape Proportion effectively characterize high-dimensional shapes.

Metrics applied to various shapes, including n-balls, demonstrate their discriminative power.

Application to the Iris dataset shows practical utility in multidimensional data analysis.

Abstract

Shape metrics for objects in high dimensions remain sparse. Those that do exist, such as hyper-volume, remain limited to objects that are better understood such as Platonic solids and $n$-Cubes. Further, understanding objects of ill-defined shapes in higher dimensions is ambiguous at best. Past work does not provide a single number to give a qualitative understanding of an object. For example, the eigenvalues from principal component analysis results in $n$ metrics to describe the shape of an object. Therefore, we need a single number which can discriminate objects with different shape from one another. Previous work has developed shape metrics for specific dimensions such as two or three dimensions. However, there is an opportunity to develop metrics for any desired dimension. To that end, we present two new shape metrics for objects in a given number of dimensions: hyper-Sphericity and hyper-Shape Proportion (SP). We explore the proprieties of these metrics on a number of different shapes including $n$-balls. We then connect these metrics to applications of analyzing the shape of multidimensional data such as the popular Iris dataset.