  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.
