Conditionally Positive Functions and p-norm Distance Matrices

TL;DR

This paper investigates the invertibility of distance matrices generated by p-norms, showing that for p in (1,2), these matrices are invertible for distinct points, but not necessarily for p > 2, impacting radial basis function interpolation.

Contribution

It extends invertibility results of Euclidean distance matrices to p-norms for p in (1,2), and demonstrates non-invertibility for p > 2 with explicit constructions.

Findings

Invertibility of p-norm distance matrices for p in (1,2)

Non-invertibility for p > 2 with explicit examples

Implications for radial basis function interpolation

Abstract

In Micchelli's paper "Interpolation of scattered data: distance matrices and conditionally positive functions", deep results were obtained concerning the invertibility of matrices arising from radial basis function interpolation. In particular, the Euclidean distance matrix was shown to be invertible for distinct data. In this paper, we investigate the invertibility of distance matrices generated by $p$-norms. In particular, we show that, for any $p\in (1, 2)$, and for distinct points $ x^1, ..., x^n \in {\cal R}^d $, where $n$ and $d$ may be any positive integers, with the proviso that $ n \ge 2$, the matrix $A \in {\cal R}^{n \times n} $ defined by $$ A_{ij} = \Vert x^i - x^j \Vert_p , \hbox{ for } 1 \le i, j \le n, $$ satisfies $$ (-1)^{n-1}\det A > 0 .$$ We also show how to construct, for every $p > 2$, a configuration of distinct points in some ${\cal R}^d$ giving a singular $p$-norm distance matrix. Thus radial basis function interpolation using $p$-norms is uniquely determined by any distinct data for $p \in (1,2]$, but not so for $p > 2$.