Rapid Mixing of Stereographic MCMC for Heavy-Tailed Sampling
Abstract
Sampling from high-dimensional, heavy-tailed distributions is a fundamental challenge in computational statistics, as many standard Markov chain Monte Carlo (MCMC) methods mix poorly in such settings. Recently, Stereographic MCMC [Yang et al., 2024] and the Sub-Cauchy Projection Sampler [Grazzi et al., 2026] have been shown to perform successfully on such tasks. However, establishing their non-asymptotic convergence properties remains an important open problem. In this work, we fill this gap by establishing non-asymptotic upper bounds on the mixing time of the stereographic projection and sub-Cauchy projection samplers. Our results demonstrate that, under certain conditions on the target and initial distributions, the mixing time is polynomial in dimension for a broad class of distributions, including light- and heavy-tailed cases. Motivated by the theoretical analysis, we further establish a new weighted isoperimetric inequality that extends the classical version for log-concave distributions to the heavy-tailed setting with optimal dimension dependence. The proof techniques provide new insights into the geometric properties of heavy-tailed distributions that govern rapid mixing in high dimensions.
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BibTeX
@misc{milinanni2026rapid,
title = {{Rapid Mixing of Stereographic MCMC for Heavy-Tailed Sampling}},
author = {Federica Milinanni and Tyler Farghly and Jun Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18513},
eprint = {2610.18513},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18513},
}
RIS
TY - UNPB
TI - Rapid Mixing of Stereographic MCMC for Heavy-Tailed Sampling
AU - Federica Milinanni
AU - Tyler Farghly
AU - Jun Yang
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18513
UR - https://arxiv.org/abs/2610.18513
AB - Sampling from high-dimensional, heavy-tailed distributions is a fundamental challenge in computational statistics, as many standard Markov chain Monte Carlo (MCMC) methods mix poorly in such settings. Recently, Stereographic MCMC [Yang et al., 2024] and the Sub-Cauchy Projection Sampler [Grazzi et al., 2026] have been shown to perform successfully on such tasks. However, establishing their non-asymptotic convergence properties remains an important open problem. In this work, we fill this gap by establishing non-asymptotic upper bounds on the mixing time of the stereographic projection and sub-Cauchy projection samplers. Our results demonstrate that, under certain conditions on the target and initial distributions, the mixing time is polynomial in dimension for a broad class of distributions, including light- and heavy-tailed cases. Motivated by the theoretical analysis, we further establish a new weighted isoperimetric inequality that extends the classical version for log-concave distributions to the heavy-tailed setting with optimal dimension dependence. The proof techniques provide new insights into the geometric properties of heavy-tailed distributions that govern rapid mixing in high dimensions.
ER -