Abstract

This paper considers the problem of regressing data points on the Grassmann manifold over a scalar-valued variable. The Grassman- nian has recently gained considerable attention in the vision community with applications in domain adaptation, face recognition, shape analy- sis, or the classification of linear dynamical systems. Motivated by the success of these approaches, we introduce a principled formulation for regression tasks on that manifold. We propose an intrinsic geodesic re- gression model generalizing classical linear least-squares regression. Since geodesics are parametrized by a starting point and a velocity vector, the model enables the synthesis of new observations on the manifold. To ex- emplify our approach, we demonstrate its applicability on three vision problems where data ob jects can be represented as points on the Grass- mannian: the prediction of traffic speed and crowd counts from dynamical system models of surveillance videos and the modeling of aging trends in human brain structures using an affine-invariant shape representation.