On the Schoenberg Transformations in Data Analysis: Theory and Illustrations

TL;DR

This paper explores Schoenberg transformations that embed Euclidean distances into higher dimensions, presenting new theoretical insights and visualizations, and demonstrating their application in a distance-based discriminant algorithm linked to Gaussian kernels.

Contribution

It introduces original results on the geometric properties of Schoenberg transformations and connects them to machine learning techniques like Gaussian kernels.

Findings

Derived new theorems on arc lengths, angles, and curvature of transformations

Visualized transformations on artificial datasets using multidimensional scaling

Demonstrated the application of transformations in a discriminant algorithm

Abstract

The class of Schoenberg transformations, embedding Euclidean distances into higher dimensional Euclidean spaces, is presented, and derived from theorems on positive definite and conditionally negative definite matrices. Original results on the arc lengths, angles and curvature of the transformations are proposed, and visualized on artificial data sets by classical multidimensional scaling. A simple distance-based discriminant algorithm illustrates the theory, intimately connected to the Gaussian kernels of Machine Learning.