Maximum volume coordinates for Grassmann interpolation: Lagrange, Hermite, and errors

Abstract

We present a novel approach to Riemannian interpolation on the Grassmann manifold. Instead of relying on the Riemannian normal coordinates, i.e. the Riemannian exponential and logarithm maps, we approach the interpolation problem with an alternative set of local coordinates and corresponding parameterizations. A special property of these coordinates is that their calculation does not require any matrix decompositions. This is a numerical advantage over Riemann normal coordinates and many other retractions on the Grassmann manifold, especially when derivative data are to be treated. To estimate the interpolation error, we examine the conditioning of these mappings and state explicit bounds. It turns out that the parameterizations are well-conditioned, but the coordinate mappings are generally not. As a remedy, we introduce maximum-volume coordinates that are based on a search for subblocks of column-orthogonal matrices of large absolute determinant. We show that the order of magnitude of the asymptotic interpolation error on $\Gr(n,p)$ is the same as in the Euclidean space. Two numerical experiments are conducted. The first is an academic one, where we interpolate a parametric orthogonal projector $QQ^T$, where the $Q$--factor stems from a parametric compact QR--decomposition. The second experiment is in the context of parametric model reduction of dynamical systems, where we interpolate reduced subspaces that are obtained by proper orthogonal decomposition.