Numerical Differential Geometry
Speaker: Lek-Heng Lim, University of Chicago
Location: Warren Weaver Hall 1302
Date: Monday, May 6, 2019, 3:45 p.m.
This is a subject that started from a classic paper of Edelman, Arias, and Smith (EAS). One of its key insight is that certain Riemannian manifolds may be given matrix coordinates and optimization algorithms on these "matrix manifolds" then require only standard numerical linear algebra, i.e., no numerical solutions of differential or nonlinear algebraic equations needed. The EAS line of work has thus far been limited to essentially three spaces: the Stiefel manifold, the Grassmannian, and the PSD cone, equipped with their natural Riemannian metrics. We will extend it to other spaces: affine Grassmannian, flag manifolds, pseudospheres, pseudohyperbolic spaces, de Sitter and anti de Sitter spaces, indefinite Stiefel and Grassmmann manifolds, indefinite Lie groups; apart from the first two, the rest are semi-Riemmannian manifolds. We will also introduce a notion of "matrix fiber bundle" -- one whose fiber, base, and total spaces are all matrix manifolds. We will see that standard matrix decompositions -- LU, QR, SVD, EVD, polar, Cholesky, Jordan, Kronecker, Schur, etc -- may all be regarded as matrix fiber bundles and how this perspective gives us new notions of distances between various matrices. This talk is based on various joint works with Ke Ye, Tingran Gao, and Ken Sze-Wai Wong.