Optimal interpolants on Grassmann manifolds

The Grassmann manifoldGrm(Rn) of all m-dimensional subspaces of the n-dimensional space Rn(m<n) is widely used in image analysis, statistics and optimization. Motivated by interpolation in the manifold Gr2(R4), we first formulate the differential equation for desired interpolation curves called Riemannian cubics in symmetric spaces by the Pontryagin maximum principle (PMP) and then narrow down to it in Gr2(R4). Although computation on this low-dimensional manifold may not occur heavy burden for modern machines, theoretical analysis for Riemannian cubics is very limited in references due to its highly nonlinearity. This paper focuses on presenting analytical and geometrical structures for the so-called Lie quadratics associated with Riemannian cubics. By analysing asymptotics of Lie quadratics, we find asymptotics of Riemannian cubics in Gr2(R4). Finally, we illustrate our results by numerical simulations.