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A Generalized Lyapunov Feature for Dynamical Systems on Riemannian Manifolds

Dynamic phenomena such as human activities, dynamic scenes, and moving crowds are commonly observed through visual sensors, resulting in feature trajectories sampled in time. Such phenomena can be accurately modeled by taking the temporal variations and changes into account. For problems where the trajectories are sufficiently different, elastic metrics can provide distances that are invariant to speed, but for more complex problems such as fine grained activity classification, one needs to exploit higher order dynamical properties. For features in the Euclidean space, applications such as crowd monitoring, dynamic scene recognition and human movement quality analysis have found a lot of success this way. In this paper we propose the largest Riemannian Lyapunov exponent (L-RLE), which is the first generalization of the largest Lyapunov exponent to Riemannian manifolds. The largest Lyapunov exponent is a classic measure to quantify the amount of chaos within signals in the Euclidean space, and allows us to exploit higher order dynamics for various applications. We show the effectiveness of the L-RLE on two manifolds the Grassmann and the SO(3) lie group. By modeling human actions as dynamic processes evolving on Riemannian manifolds, we show that L-RLE can measure the amount of chaos within each action accurately. We show that our measure is a good generalization of largest Euclidean Lyapunov exponent (L-ELE), and is less susceptible to arbitrary distortions.