TY - GEN
T1 - The Classical and Symplectic Stiefel and Grassmann Manifolds
T2 - Geometry and Applications
AU - Bendokat, Thomas
PY - 2021/11/30
Y1 - 2021/11/30
N2 - Geodesics, the related exponential map, and its inverse, or approximations of the latter two, lie at the heart of most data processing operations on matrix manifolds. When (efficient) formulas are available, it becomes possible to tackle tasks such as interpolation or optimization problems. This thesis solves related problems on the classical and symplectic Stiefel and Grassmann manifolds, i.e., the manifolds of linear or symplectic bases or subspaces of vector spaces, respectively, where the latter---the symplectic Grassmann manifold---is introduced as a new object of study in this thesis.For the classical Grassmann manifold, an algorithm to compute the Riemannian logarithm is derived, which is numerically efficient and, as a new result, allows the computation of geodesics between any two given points. Furthermore, research tracks treating the Grassmann manifold as a set of projectors and as a quotient of the Stiefel manifold are combined and formulas for an easy transition between the two are stated. At last, a full description of the conjugate locus is found.For the classical Stiefel manifold, an efficient method to compute a known kind of quasi-geodesics between two given points is found and an alternative kind of connecting quasi-geodesics, much closer to the Riemannian geodesics, is introduced.For the symplectic Stiefel manifold, a pseudo-Riemannian and Riemannian framework are introduced, which allow for the computation of (for any metric hitherto unknown) geodesics. For the newly introduced symplectic Grassmann manifold, pseudo-Riemannian and Riemannian metrics and corresponding geodesics, with efficient formulas via horizontal lifts, are found in a similar fashion. On both the symplectic Stiefel and Grassmann manifold, efficient formulas to compute and invert the Cayley retraction are introduced.
AB - Geodesics, the related exponential map, and its inverse, or approximations of the latter two, lie at the heart of most data processing operations on matrix manifolds. When (efficient) formulas are available, it becomes possible to tackle tasks such as interpolation or optimization problems. This thesis solves related problems on the classical and symplectic Stiefel and Grassmann manifolds, i.e., the manifolds of linear or symplectic bases or subspaces of vector spaces, respectively, where the latter---the symplectic Grassmann manifold---is introduced as a new object of study in this thesis.For the classical Grassmann manifold, an algorithm to compute the Riemannian logarithm is derived, which is numerically efficient and, as a new result, allows the computation of geodesics between any two given points. Furthermore, research tracks treating the Grassmann manifold as a set of projectors and as a quotient of the Stiefel manifold are combined and formulas for an easy transition between the two are stated. At last, a full description of the conjugate locus is found.For the classical Stiefel manifold, an efficient method to compute a known kind of quasi-geodesics between two given points is found and an alternative kind of connecting quasi-geodesics, much closer to the Riemannian geodesics, is introduced.For the symplectic Stiefel manifold, a pseudo-Riemannian and Riemannian framework are introduced, which allow for the computation of (for any metric hitherto unknown) geodesics. For the newly introduced symplectic Grassmann manifold, pseudo-Riemannian and Riemannian metrics and corresponding geodesics, with efficient formulas via horizontal lifts, are found in a similar fashion. On both the symplectic Stiefel and Grassmann manifold, efficient formulas to compute and invert the Cayley retraction are introduced.
U2 - 10.21996/mmer-zp86
DO - 10.21996/mmer-zp86
M3 - Ph.D. thesis
PB - Syddansk Universitet. Det Naturvidenskabelige Fakultet
ER -