Characteristics of Distance Matrices, the Second Look

TL;DR

This paper provides a rigorous mathematical analysis of distance matrices, exploring their limits and properties as data dimensions grow, with a focus on concepts like robustness and correlation, supported by illustrative examples.

Contribution

It offers a precise mathematical characterization of distance matrix limits and examines robustness and correlation definitions in a purely mathematical context.

Findings

Limits of functions of distance matrices are characterized as data dimensions increase.

Robustness can be zero for small data matrices under certain definitions.

Examples demonstrate the behavior of robustness and correlation in low-dimensional cases.

Abstract

Here the definitions of nearest neighbor, robustness, concordance, and correlation, all of which feature in (Temple 2023) (henceforth abbreviated (T23)), are adjusted to make them completely mathematical while preserving their significance. A characterization is given of the possible limits of a function of distance matrices as the data matrices from which they are derived acquire more and more columns while their number of rows and the distance defining norm (= coefficient, in the terminology of (T23)) are held fixed. No data contribute to the discussion here, but many examples, with standard norms and data matrices having just a few rows and columns, play an important role. Indeed, small data matrices are displayed showing that robustness, defined either of the two ways, can be zero.