# Local and Global Perspectives on Diffusion Maps in the Analysis of Molecular Systems

## TL;DR

This paper explores local and global diffusion map techniques to analyze molecular systems, identifying metastable states and accelerating sampling in high-dimensional molecular dynamics simulations.

## Contribution

It introduces a unified framework contrasting local and global diffusion map perspectives and proposes an enhanced sampling algorithm for molecular systems.

## Key findings

- Diffusion maps effectively identify metastable states in molecular dynamics.
- The global approach approximates transition probabilities and committor functions.
- The local approach converges within quasi-stationary distributions and improves sampling efficiency.

## Abstract

Diffusion maps approximate the generator of Langevin dynamics from simulation data. They afford a means of identifying the slowly-evolving principal modes of high-dimensional molecular systems. When combined with a biasing mechanism, diffusion maps can accelerate the sampling of the stationary Boltzmann-Gibbs distribution. In this work, we contrast the local and global perspectives on diffusion maps, based on whether or not the data distribution has been fully explored. In the global setting, we use diffusion maps to identify metastable sets and to approximate the corresponding committor functions of transitions between them. We also discuss the use of diffusion maps within the metastable sets, formalising the locality via the concept of the quasi-stationary distribution and justifying the convergence of diffusion maps within a local equilibrium. This perspective allows us to propose an enhanced sampling algorithm. We demonstrate the practical relevance of these approaches both for simple models and for molecular dynamics problems (alanine dipeptide and deca-alanine).