# Stereographic Markov Chain Monte Carlo

arXiv: https://arxiv.org/abs/2205.12112 · HTML: https://mlsanity.com/paper/2205.12112/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 42 references that are in the map.
- First posted on arXiv 24 May 2022.

## Builds on (most cited first, 5 of 5)

- [The Bouncy Particle Sampler: A Non-Reversible Rejection-Free Markov Chain Monte Carlo Method](https://mlsanity.com/paper/1510.02451/) (2015)
- [A Framework for Adaptive MCMC Targeting Multimodal Distributions](https://mlsanity.com/paper/1812.02609/) (2018)
- [Piecewise Deterministic Markov Processes for Continuous-Time Monte Carlo](https://mlsanity.com/paper/1611.07873/) (2016)
- [Rapid Mixing of Geodesic Walks on Manifolds with Positive Curvature](https://mlsanity.com/paper/1609.02901/) (2016)
- [Efficient strategy for the Markov chain Monte Carlo in high-dimension with heavy-tailed target probability distribution](https://mlsanity.com/paper/1412.6231/) (2014)

## Built on by (most cited first, 6 of 6)

- [Towards a Complete Analysis of Langevin Monte Carlo: Beyond Poincar'e Inequality](https://mlsanity.com/paper/2303.03589/) (2023)
- [Flexible Tails for Normalizing Flows](https://mlsanity.com/paper/2406.16971/) (2024)
- [A Separation in Heavy-Tailed Sampling: Gaussian vs. Stable Oracles for Proximal Samplers](https://mlsanity.com/paper/2405.16736/) (2024)
- [Mean-Square Analysis of Discretized Ito Diffusions for Heavy-tailed Sampling](https://mlsanity.com/paper/2303.00570/) (2023)
- [Piecewise deterministic generative models](https://mlsanity.com/paper/2407.19448/) (2024)
- [From Minimax Optimal Importance Sampling to Uniformly Ergodic Importance-tempered MCMC](https://mlsanity.com/paper/2506.19186/) (2025)

## Abstract (arXiv)

High-dimensional distributions, especially those with heavy tails, are notoriously difficult for off-the-shelf MCMC samplers: the combination of unbounded state spaces, diminishing gradient information, and local moves results in empirically observed ``stickiness'' and poor theoretical mixing properties -- lack of geometric ergodicity. In this paper, we introduce a new class of MCMC samplers that map the original high-dimensional problem in Euclidean space onto a sphere and remedy these notorious mixing problems. In particular, we develop random-walk Metropolis type algorithms as well as versions of the Bouncy Particle Sampler that are uniformly ergodic for a large class of light and heavy-tailed distributions and also empirically exhibit rapid convergence in high dimensions. In the best scenario, the proposed samplers can enjoy the ``blessings of dimensionality'' that the convergence is faster in higher dimensions.