Parametric Regression on the Grassmannian

TL;DR

This paper introduces a novel method for intrinsic parametric regression on the Grassmann manifold, extending linear models to handle complex data like shapes and videos with simple, unified solutions.

Contribution

It generalizes Euclidean least-squares to Riemannian manifolds, providing a flexible framework for regression on the Grassmannian, including geodesics, time-warped models, and splines.

Findings

Effective shape regression as a function of age

Accurate traffic-speed estimation from video data

Reliable crowd-counting in surveillance videos

Abstract

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.