Simpler flag optimization

TL;DR

This paper explores the geometry of flag manifolds under various embeddings, providing explicit formulas for geometric objects and developing an efficient coordinate descent method that outperforms existing gradient methods.

Contribution

It introduces closed-form expressions for geometric operations on flag manifolds and proposes a novel coordinate descent algorithm with superior performance.

Findings

Explicit formulas for tangent vectors, metrics, and geodesics on flag manifolds.

A new coordinate descent method tailored for flag manifolds.

Demonstrated improved convergence over traditional gradient descent methods.

Abstract

We study the geometry of flag manifolds under different embeddings into a product of Grassmannians. We show that differential geometric objects and operations -- tangent vector, metric, normal vector, exponential map, geodesic, parallel transport, gradient, Hessian, etc -- have closed-form analytic expressions that are computable with standard numerical linear algebra. Furthermore, we are able to derive a coordinate descent method in the flag manifold that performs well compared to other gradient descent methods.