Comments on "On Approximating Euclidean Metrics by Weighted t-Cost Distances in Arbitrary Dimension"

TL;DR

This paper critically evaluates Mukherjee’s weighted t-cost distances as Euclidean norm approximations, revealing optimistic error estimates and proposing a normalization method that enhances accuracy in high-dimensional spaces.

Contribution

It compares existing Euclidean approximation methods, identifies inaccuracies in previous error estimates, and introduces a normalization scheme that improves approximation accuracy.

Findings

Mukherjee's error estimates are overly optimistic.

Normalization significantly reduces approximation errors.

Proposed method improves accuracy in high-dimensional spaces.

Abstract

Mukherjee (Pattern Recognition Letters, vol. 32, pp. 824-831, 2011) recently introduced a class of distance functions called weighted t-cost distances that generalize m-neighbor, octagonal, and t-cost distances. He proved that weighted t-cost distances form a family of metrics and derived an approximation for the Euclidean norm in $\mathbb{Z}^n$. In this note we compare this approximation to two previously proposed Euclidean norm approximations and demonstrate that the empirical average errors given by Mukherjee are significantly optimistic in $\mathbb{R}^n$. We also propose a simple normalization scheme that improves the accuracy of his approximation substantially with respect to both average and maximum relative errors.