Direct minimization of electronic structure calculations with Householder reflections

TL;DR

This paper introduces a Householder reflection-based minimization method for electronic structure calculations on the Grassman manifold, improving convergence rates by incorporating manifold geometry into the optimization process.

Contribution

It presents a novel Householder transform approach for optimization on the Grassman manifold with asymptotic complexity mn^2, enhancing convergence over traditional methods.

Findings

Significantly improved convergence rates with manifold-aware optimization.

Efficient Householder-based method with asymptotic complexity mn^2.

Comparison shows superiority over projected nonlinear conjugate gradient.

Abstract

We consider a minimization scheme based on the Householder transport operator for the Grassman manifold, where a point on the manifold is represented by a m x n matrix with orthonormal columns. In particular, we consider the case where m >> n and present a method with asymptotic complexity mn^2. To avoid explicit parametrization of the manifold we use Householder transforms to move on the manifold, and present a formulation for simultaneous Householder reflections for S-orthonormal columns. We compare a quasi-Newton and nonlinear conjugate gradient implementation adapted to the manifold with a projected nonlinear conjugate gradient method, and demonstrate that the convergence rate is significantly improved if the manifold is taken into account when designing the optimization procedure.