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{ö}m sampling. Empirical experiments support the merits of DID.

We consider the problem of localization and navigation of Autonomous Underwater Vehicles (AUV) in the context of high performance subsea asset inspection missions in deep water. We propose a solution based on the recently introduced Unscented Kalman Filter on Manifolds (UKF-M) for onboard navigation to estimate the robot's location, attitude and velocity, using a precise round and rotating Earth navigation model. Our algorithm has the merit of seamlessly handling nonlinearity of attitude, and is far simpler to implement than the extended Kalman filter (EKF), which is widely used in the navigation industry. The unscented transform notably spares the user the computation of Jacobians and lends itself well to fast prototyping in the context of multi-sensor data fusion. Besides, we provide the community with feedback about implementation, and execution time is shown to be compatible with real-time. Realistic extensive Monte-Carlo simulations prove uncertainty is estimated with accuracy by the filter, and illustrate its convergence ability. Real experiments in the context of a 900m deep dive near Marseille (France) illustrate the relevance of the method.