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Parametric Regression on the Grassmannian.

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We address the problem of fitting parametric curves on the Grassmann manifold for the purpose of intrinsic parametric regression. We start from the energy minimization formulation of linear least-squares in Euclidean space and generalize this concept to general nonflat Riemannian manifolds, following an optimal-control point of view. We then specialize this idea to the Grassmann manifold and demonstrate that it yields a simple, extensible and easy-to-implement solution to the parametric regression problem. In fact, it allows us to extend the basic geodesic model to (1) a "time-warped" variant and (2) cubic splines. We demonstrate the utility of the proposed solution on different vision problems, such as shape regression as a function of age, traffic-speed estimation and crowd-counting from surveillance video clips. Most notably, these problems can be conveniently solved within the same framework without any specifically-tailored steps along the processing pipeline.

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