Dimension Walks on Generalized Spaces

TL;DR

This paper introduces linear operators that enable dimension walks on generalized spaces, specifically on products of spheres and Euclidean spaces, while maintaining positive definiteness of associated functions.

Contribution

It proposes novel linear operators for dimension walks on generalized spaces, extending the theory of positive definite functions to these complex structures.

Findings

Operators preserve positive definiteness during dimension walks

Extension of positive definite functions to product spaces

Framework applicable to high-dimensional data analysis

Abstract

Let $d,k$ be positive integers. We call generalized spaces the cartesian product of the $d$-dimensional sphere, $\mathbb{S}^d$, with the $k$-dimensional Euclidean space, $\mathbb{R}^k$. We consider the class ${\mathcal P}(\mathbb{S}^d \times \mathbb{R}^k)$ of continuous functions $\varphi: [-1,1] \times [0,\infty) \to \mathbb{R}$ such that the mapping $C: \left ( \mathbb{S}^d \times\mathbb{R}^k \right )^2 \to \mathbb{R}$, defined as $C \Big ( (x,y),(x^{\prime},y^{\prime})\Big ) = \varphi \Big ( \cos \theta(x,x^{\prime}), \|y-y^{\prime}\| \Big )$, $(x,y), \; (x^{\prime},y^{\prime}) \in \mathbb{S}^d \times \mathbb{R}^k$, is positive definite. We propose linear operators that allow for walks through dimension within generalized spaces while preserving positive definiteness.