Factorization of the Determinant of the Gaussian-Correlation Matrix of Evenly Spaced Points Using an Inter-dimensional Multiset Duality

TL;DR

This paper proves a formula for the determinant of a Gaussian-correlation matrix of evenly spaced points, revealing its dependence on the nearest-neighbor distance, and introduces a novel inter-dimensional multiset duality during the factorization.

Contribution

It provides a new factorization of the determinant of Gaussian-correlation matrices using Neville elimination and introduces an inter-dimensional multiset duality concept.

Findings

Determinant has leading power n(n-1) in nearest-neighbor distance.

Neville elimination yields all elements of the upper triangular matrix U.

Conjecture that the matrix V is strictly totally positive.

Abstract

We prove that the determinant of a Gaussian-correlation matrix V of n evenly spaced points has leading power n(n-1) in the nearest-neighbor distance between points. The proof uses Neville elimination to determine all elements of the upper triangular matrix U of V and provides a factorization of det(V). The proof makes use of an inter-dimensional multiset duality involving simplices that emerge during the factorization. We conjecture that V for evenly spaced points is strictly totally positive.