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

TL;DR

This paper introduces a simplified approach to Riemannian submanifold optimization with momentum, making it more computationally feasible for structured positive-definite matrices and deep learning applications.

Contribution

It proposes a generalized Riemannian normal coordinate system that simplifies optimization on structured positive-definite matrices, enabling matrix-inverse-free second-order methods for deep learning.

Findings

Simplifies Riemannian submanifold optimization with momentum.

Enables matrix-inverse-free second-order optimization in deep learning.

Provides a practical implementation with code available online.

Abstract

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 positive-definite 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 $2^\text{nd}$-order optimizers for deep learning with low precision by using only matrix multiplications. Code: https://github.com/yorkerlin/StructuredNGD-DL