Parametric Regression on the Grassmannian

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.Comment: 14 pages, 11 figure

Discrete regression methods on the cone of positive-definite matrices

We consider the problem of fitting a discrete curve to time-labeled data points on the set P n of all n-by-n symmetric positive-definite matrices. The quality of a curve is measured by a weighted sum of a term that penalizes its lack of fit to the data and a regularization term that penalizes speed and acceleration. The corresponding objective function depends on the choice of a Riemannian metric on P n. We consider the Euclidean metric, the Log-Euclidean metric and the affine-invariant metric. For each, we derive a numerical algorithm to minimize the objective function. We compare these in terms of reliability and speed, and we assess the visual appear ance of the solutions on examples for n = 2. Notably, we find that the Log-Euclidean and the affine-invariant metrics tend to yield similar-and sometimes identical-results, while the former allows for much faster and more reliable algorithms than the latter