Parametric Regression on the Grassmannian
This work provides a general framework for parametric regression on nonflat manifolds, applicable to vision problems like shape regression and traffic-speed estimation, but it is incremental as it builds on existing energy minimization formulations.
The paper tackles the problem of fitting parametric curves on the Grassmann manifold for intrinsic parametric regression, resulting in a simple, extensible, and easy-to-implement solution that extends basic geodesic models to time-warped variants and cubic splines.
We address the problem of fitting parametric curves on the Grassmann manifold for the purpose of intrinsic parametric regression. As customary in the literature, we start from the energy minimization formulation of linear least-squares in Euclidean spaces 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.