  Robust representation learning of temporal dynamic interactions is an
important problem in robotic learning in general and automated unsupervised
learning in particular. Temporal dynamic interactions can be described by
(multiple) geometric trajectories in a suitable space over which unsupervised
learning techniques may be applied to extract useful features from raw and
high-dimensional data measurements. Taking a geometric approach to robust
representation learning for temporal dynamic interactions, it is necessary to
develop suitable metrics and a systematic methodology for comparison and for
assessing the stability of an unsupervised learning method with respect to its
tuning parameters. Such metrics must account for the (geometric) constraints in
the physical world as well as the uncertainty associated with the learned
patterns. In this paper we introduce a model-free metric based on the
Procrustes distance for robust representation learning of interactions, and an
optimal transport based distance metric for comparing between distributions of
interaction primitives. These distance metrics can serve as an objective for
assessing the stability of an interaction learning algorithm. They are also
used for comparing the outcomes produced by different algorithms. Moreover,
they may also be adopted as an objective function to obtain clusters and
representative interaction primitives. These concepts and techniques will be
introduced, along with mathematical properties, while their usefulness will be
demonstrated in unsupervised learning of vehicle-to-vechicle interactions
extracted from the Safety Pilot database, the world's largest database for
connected vehicles.
