Learning pullback manifolds of dynamical models

In this paper we present a general differential-geometric framework for learning Riemannian metrics or distance functions for dynamical models, given a training set which can be either labeled or unlabeled. Given a training set of models, the optimal metric is selected among a family of pullback metrics induced by a parameterized automorphism of the space of models. The problem of classifying motions, encoded as dynamical models of a certain class, can then be posed on the learnt manifold. As significant case studies, in virtue of their applicability to gait identification and action recognition, we consider the class of multidimensional autoregressive models of order 2 and that of hidden Markov models. We study their manifolds and design automorphisms there which allow to build parametric families of metrics we can optimize upon. Experimental results concerning action and identity recognition are presented, which show how such optimal pullback Fisher metrics greatly improve classification performances. F. Cuzzolin is Lecturer and Early Career Fellow with the Department of Computing, Oxford Brookes University, Oxford, United Kingdom. URL: http://cms.brookes.ac.uk/staff/FabioCuzzolin/. E-mail: [email protected] Phone: +44 (0)1865 484526. IEEE TRANSACTIONS ON PAMI, VOL. XX, NO. Y, MONTH 2010 2