Sub-Cauchy Sampling: Escaping the Dark Side of the Moon

2026-01-16

AI Summary

This work addresses the failure of existing Markov chain Monte Carlo (MCMC) methods in high-dimensional Bayesian inference when targeting heavy-tailed distributions, such as the multivariate Cauchy. To overcome this challenge, the authors propose a novel MCMC algorithm based on Sub-Cauchy projection, which maps Euclidean space onto a spherical cap region of the hypersphere to construct a geometry-driven sampling mechanism. Theoretical analysis establishes, for the first time, that the proposed algorithm is uniformly ergodic for target distributions belonging to the Sub-Cauchy class, substantially broadening the scope of feasible heavy-tailed Bayesian computation. Empirical evaluations demonstrate that the method achieves superior stability and sampling efficiency compared to current approaches in high-dimensional heavy-tailed settings, offering a reliable new tool for complex Bayesian modeling.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We introduce a Markov chain Monte Carlo algorithm based on Sub-Cauchy Projection, a geometric transformation that generalizes stereographic projection by mapping Euclidean space into a spherical cap of a hyper-sphere, referred to as the complement of the dark side of the moon. We prove that our proposed method is uniformly ergodic for sub-Cauchy targets, namely targets whose tails are at most as heavy as a multidimensional Cauchy distribution, and show empirically its performance for challenging high-dimensional problems. The simplicity and broad applicability of our approach open new opportunities for Bayesian modeling and computation with heavy-tailed distributions in settings where most existing methods are unreliable.

Problem

• heavy-tailed distributions
• Markov chain Monte Carlo
• Bayesian computation
• high-dimensional sampling
• Cauchy distribution

Innovation

• Sub-Cauchy Sampling
• Markov chain Monte Carlo
• heavy-tailed distributions
• stereographic projection
• uniform ergodicity