Measuring dissimilarity with diffeomorphism invariance

TL;DR

This paper introduces DID, a novel dissimilarity measure invariant to diffeomorphisms, leveraging data structure for improved similarity assessment in machine learning.

Contribution

The paper presents DID, a new invariant dissimilarity measure applicable to various data spaces, with a theoretical foundation and practical approximation methods.

Findings

DID is invariant to diffeomorphisms and suitable for diverse data types.

DID can be computed efficiently using Nyström sampling.

Empirical results demonstrate DID's effectiveness in real data scenarios.

Abstract

Measures of similarity (or dissimilarity) are a key ingredient to many machine learning algorithms. We introduce DID, a pairwise dissimilarity measure applicable to a wide range of data spaces, which leverages the data's internal structure to be invariant to diffeomorphisms. We prove that DID enjoys properties which make it relevant for theoretical study and practical use. By representing each datum as a function, DID is defined as the solution to an optimization problem in a Reproducing Kernel Hilbert Space and can be expressed in closed-form. In practice, it can be efficiently approximated via Nystr\"om sampling. Empirical experiments support the merits of DID.