During the last phase of my PhD I worked together with Dr. Payel Das, Dr. Simon Hadfield, Dr. Yunpeng Li and Dr. Paula Gherghinescu on a project aimed at accelerating Hamiltonian Monte-Carlo. This project was quite a cross-disciplinary effort, as we had to combine ideas from (astro)physics and geometric mechanics together with computer science and probabilistic mathematics.
In this study we include results looking at the trade-off between the map-building and the faster sampling, but the techniques to build these maps are in rapid development, and might get more efficient soon.
The process of this research project has been a profound experience for me. I think because, for both me and Payel, the research topics covered in this project have generally been new to both of us. Over the entire project i've been on a journey from learning contemporary machine learning technique and probabilistic programming, to galactic dynamics, to transport theory, and finally to geometric mechanics. Awesome stuff.
Hamiltonian Monte-Carlo is a Monte-Carlo sampling/inference technique which uses techniques from classical mechanics to guide the sampling through the posterior. This technique is efficient in high dimensions, and is generally faster than the standard Metropolis-Hastings in uncovering the posterior landscape.
The core idea of this project is to make use of several transformations. The first transformation is to map the target distribution (posterior) to a known, controlled, base distribution, and the second transformation is a transformation to action-angle space. Combining these two transformations enables a fast sampling of (complex) posterior shapes, but there is a price to pay. Learning the transformation between the target and the base distribution will require computation.
We are currently wrapping up the project and aim to publish the method and some use-case studies in an astrophysics journal. When the paper is submitted I will write down a more detailed version of the project here.
Recently we applied for a low-TRL early stage research and development scheme grant (https://www.ukri.org/opportunity/early-stage-research-and-development-scheme/) to be able to continue this research and the development and application of the AAHMC sampling.