Intrinsic representation of tangent vectors and vector transports on matrix manifolds

Abstract

The quasi-Newton methods on Riemannian manifolds proposed thus far do not appear to lend themselves to satisfactory convergence analyses unless they resort to an isometric vector transport. This prompts us to propose a computationally tractable isometric vector transport on the Stiefel manifold of orthonormal p-frames in $\mathbb {R}^n$. Specifically, it requires $O(np^2)$ flops, which is considerably less expensive than existing alternatives in the frequently encountered case where $n\gg p$. We then build on this result to also propose computationally tractable isometric vector transports on other manifolds, namely the Grassmann manifold, the fixed-rank manifold, and the positive-semidefinite fixed-rank manifold. In the process, we also propose a convenient way to represent tangent vectors to these manifolds as elements of $\mathbb {R}^d$, where d is the dimension of the manifold. We call this an “intrinsic” representation, as opposed to “extrinsic” representations as elements of $\mathbb {R}^w$, where w is the dimension of the embedding space. Finally, we demonstrate the performance of the proposed isometric vector transport in the context of a Riemannian quasi-Newton method applied to minimizing the Brockett cost function.