  Visual observations of dynamic phenomena, such as human actions, are often
represented as sequences of smoothly-varying features . In cases where the
feature spaces can be structured as Riemannian manifolds, the corresponding
representations become trajectories on manifolds. Analysis of these
trajectories is challenging due to non-linearity of underlying spaces and
high-dimensionality of trajectories. In vision problems, given the nature of
physical systems involved, these phenomena are better characterized on a
low-dimensional manifold compared to the space of Riemannian trajectories. For
instance, if one does not impose physical constraints of the human body, in
data involving human action analysis, the resulting representation space will
have highly redundant features. Learning an effective, low-dimensional
embedding for action representations will have a huge impact in the areas of
search and retrieval, visualization, learning, and recognition. The difficulty
lies in inherent non-linearity of the domain and temporal variability of
actions that can distort any traditional metric between trajectories. To
overcome these issues, we use the framework based on transported square-root
velocity fields (TSRVF); this framework has several desirable properties,
including a rate-invariant metric and vector space representations. We propose
to learn an embedding such that each action trajectory is mapped to a single
point in a low-dimensional Euclidean space, and the trajectories that differ
only in temporal rates map to the same point. We utilize the TSRVF
representation, and accompanying statistical summaries of Riemannian
trajectories, to extend existing coding methods such as PCA, KSVD and Label
Consistent KSVD to Riemannian trajectories or more generally to Riemannian
functions.
