Lipschitz Continuity of Mahalanobis Distances and Bilinear Forms

TL;DR

This paper establishes the Lipschitz continuity of Mahalanobis distances and bilinear forms, providing tight bounds that are crucial for theoretical guarantees in machine learning models.

Contribution

It is the first to formally prove Lipschitz continuity of Mahalanobis distances and derive tight Lipschitz constants for these metrics.

Findings

Mahalanobis distance is Lipschitz continuous.

Tight Lipschitz constants for Mahalanobis distances and bilinear forms are derived.

First formal proof of Lipschitz continuity for Mahalanobis distance.

Abstract

Many theoretical results in the machine learning domain stand only for functions that are Lipschitz continuous. Lipschitz continuity is a strong form of continuity that linearly bounds the variations of a function. In this paper, we derive tight Lipschitz constants for two families of metrics: Mahalanobis distances and bounded-space bilinear forms. To our knowledge, this is the first time the Mahalanobis distance is formally proved to be Lipschitz continuous and that such tight Lipschitz constants are derived.