# Accelerated Optimization With Orthogonality Constraints

## TL;DR

This paper introduces a generalized accelerated gradient method tailored for optimization problems with orthogonality constraints, demonstrating improved iteration scaling and performance on large, ill-conditioned problems.

## Contribution

It presents a novel accelerated gradient method specifically designed for orthogonality-constrained optimization, outperforming existing quasi-Newton methods.

## Key findings

- Iteration count scales with the square root of the condition number.
- The method outperforms state-of-the-art quasi-Newton methods on large, ill-conditioned problems.
- Numerical experiments validate the efficiency and effectiveness of the proposed approach.

## Abstract

We develop a generalization of Nesterov's accelerated gradient descent method which is designed to deal with orthogonality constraints. To demonstrate the effectiveness of our method, we perform numerical experiments which demonstrate that the number of iterations scales with the square root of the condition number, and also compare with existing state-of-the-art quasi-Newton methods on the Stiefel manifold. Our experiments show that our method outperforms existing state-of-the-art quasi-Newton methods on some large, ill-conditioned problems.