Rapid mixing for Gibbs measures in Riemannian manifolds

2026-06-12 · Source: stat.ML updates on arXiv.org · Field: Science & Research — Mathematics & Computational Sciences · Depth: Expert, quick

Summary

The paper "Rapid mixing for Gibbs measures in Riemannian manifolds," submitted on 11 Jun 2026 (arXiv:2606.13453), analyzes Langevin dynamics on Riemannian manifolds. It identifies specific conditions that ensure rapid mixing to the Gibbs measure, characterized by a suitable logarithmic Sobolev inequality. These conditions are tied to the manifold's curvature, the inverse temperature, and the presence of escaping directions from saddle points, while actively excluding barren plateaus and spurious local minima. When these criteria are met, the research demonstrates that mixing times polynomial in the manifold's dimension are achievable. This significant result is derived through establishing a novel relationship between Langevin processes operating in the domain and the image of a Riemannian submersion, a connection noted for its potential independent interest.

Key takeaway

For Research Scientists analyzing the convergence of Langevin dynamics in complex, high-dimensional spaces, this work offers critical theoretical insights. You should consider how manifold curvature, inverse temperature, and the presence of escaping saddle points directly influence mixing times and the avoidance of problematic landscapes like barren plateaus. This understanding can guide the design of more robust and efficient sampling or optimization algorithms by leveraging geometric properties.

Key insights

Conditions for rapid mixing of Langevin dynamics on Riemannian manifolds are identified, achieving polynomial mixing times.

Principles

• Manifold curvature and inverse temperature dictate mixing.
• Escaping saddle points avoids barren plateaus.
• Riemannian submersion relations aid mixing analysis.

Method

The paper uses a relation between Langevin processes in the domain and image of a Riemannian submersion to prove polynomial mixing times. This analytical approach links dynamics across different geometric spaces.

Topics

• Langevin Dynamics
• Riemannian Manifolds
• Gibbs Measures
• Rapid Mixing
• Logarithmic Sobolev Inequality
• Saddle Points

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.