Rate-invariant analysis of covariance trajectories - Zhang, Zhengwu; Su, Jingyong; Klassen, Eric; Le, Huiling; Srivastava, Anuj (Springer, 2018)
Statistical analysis of dynamic systems, such as videos and dynamic functional connectivity, is often translated into a problem of analyzing trajectories of relevant features, particularly covariance matrices. As an example, in video-based action recognition, a natural mathematical representation of activity videos is as parameterized trajectories on the set of symmetric, positive-definite matrices (SPDMs). The variable execution-rates of actions, implying arbitrary parameterizations of trajectories, complicates their analysis and classification. To handle this challenge, we represent covariance trajectories using transported square-root vector fields (TSRVFs), constructed by parallel translating scaled-velocity vectors of trajectories to their starting points. The space of such representations forms a vector bundle on the SPDM manifold. Using a natural Riemannian metric on this vector bundle, we approximate geodesic paths and geodesic distances between trajectories in the quotient space of this vector bundle. This metric is invariant to the action of the reparameterization group, and leads to a rate-invariant analysis of trajectories. In the process, we remove the parameterization variability and temporally register trajectories during analysis. We demonstrate this framework in multiple contexts, using both generative statistical models and discriminative data analysis. The latter is illustrated using several applications involving video-based action recognition and dynamic functional connectivity analysis.