Using Witten Laplacians to locate index-1 saddle points

TL;DR

This paper presents a novel stochastic algorithm leveraging Witten Laplacians to efficiently locate index-1 saddle points in high-dimensional functions, with demonstrated success on molecular systems.

Contribution

It introduces a new stochastic method based on Witten Laplacians and probabilistic PDE representations for locating saddle points, extending gradient descent techniques.

Findings

Effective in high dimensions

Successfully applied to molecular systems

Outperforms traditional methods in locating saddle points

Abstract

We introduce a new stochastic algorithm to locate the index-1 saddle points of a function $V:\mathbb R^d \to \mathbb R$, with $d$ possibly large. This algorithm can be seen as an equivalent of the stochastic gradient descent which is a natural stochastic process to locate local minima. It relies on two ingredients: (i) the concentration properties on index-1 saddle points of the first eigenmodes of the Witten Laplacian (associated with $V$) on $1$-forms and (ii) a probabilistic representation of a partial differential equation involving this differential operator. Numerical examples on simple molecular systems illustrate the efficacy of the proposed approach.