Langevin dynamics on Riemannian manifolds is analyzed. Conditions ensuring the existence of a suitable logarithmic Sobolev inequality (rapid mixing to the Gibbs measure) are identified. These conditions involve the curvature of the manifold, the inverse temperature, escaping directions from saddle points, and exclude barren plateaus and spurious local minima. We show that when these conditions are met, mixing times polynomial in the dimension of the manifold are achievable. This result is obtained through a relation between Langevin processes in the domain and in the image of a Riemannian submersion. Such a relation can be of independent interest.

翻译：分析了黎曼流形上的朗之万动力学。我们识别了确保存在合适对数索博列夫不等式（快速混合至吉布斯测度）的条件。这些条件涉及流形的曲率、逆温度、鞍点的逃逸方向，并排除了贫瘠高原与虚假局部极小值。研究表明，当这些条件满足时，可在流形维数的多项式时间内实现混合。该结果通过定义域内与黎曼淹没像中朗之万过程之间的关系获得，这一关系可能具有独立的研究价值。