Measuring dissimilarity with diffeomorphism invariance
This work addresses the need for robust similarity measures in machine learning algorithms, though it appears incremental as it builds on existing kernel methods and optimization frameworks.
The authors tackled the problem of measuring dissimilarity in machine learning by introducing DID, a pairwise dissimilarity measure that is invariant to diffeomorphisms and applicable across various data spaces, achieving efficient approximation via Nyström sampling with empirical validation.
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 Nyström sampling. Empirical experiments support the merits of DID.