Beyond Langevin: Sampling Multimodal Densities using the Witten Laplacian on 1-forms

We introduce a new algorithm to sample from a probability density $\pi \propto e^{-V}$ with $V(x): \mathbb{R}^d \rightarrow \mathbb{R}$ a strongly multi-modal potential. Such distributions are challenging to sample from using standard gradient based methods, as exploration typically relies on inefficient diffusions. Inspired by connections between Schrodinger operators, Langevin diffusions and Morse theory, we develop a particle based method that exploits fast relaxation to transition pathways of the so-called Witten PDE on 1-forms (vector fields). Once pathways joining modes are sufficiently well explored, the method relaxes to inexpensive Langevin dynamics for sampling within approximately convex basins. Numerical experiments demonstrate superior performance over cheap gradient based sampling methods and competitive performance at lower cost compared to a state of the art method, parallel tempering.