Simplifying Momentum-based Positive-definite Submanifold Optimization with Applications to Deep Learning

Riemannian submanifold optimization with momentum is computationally challenging because, to ensure that the iterates remain on the submanifold, we often need to solve difficult differential equations. Here, we simplify such difficulties for a class of sparse or structured symmetric positivedefinite matrices with the affine-invariant metric. We do so by proposing a generalized version of the Riemannian normal coordinates that dynamically orthonormalizes the metric and locally converts the problem into an unconstrained problem in the Euclidean space. We use our approach to simplify existing approaches for structured covariances and develop matrix-inverse-free 2 nd -order optimizers for deep learning with low precision by using only matrix multiplications. Because the set of SPD matrices forms a Riemannian manifold, one can use Riemannian gradient methods for SPD estimation, but this can be computationally infeasible in high-dimensions. This is because the methods often require full-rank matrix decomposition (see Table 1 ). Computations