For Gibbs measures whose optimal set is a compact submanifold S, the Poincare constant is bounded below by the first nonzero eigenvalue of the Laplace-Beltrami operator on S, for low temperature.
High-dimensional limit theorems for sgd: Effective dynamics and critical scaling
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Poincare Inequality for Local Log-Polyak-\L ojasiewicz Measures: Non-asymptotic Analysis in Low-temperature Regime
For Gibbs measures whose optimal set is a compact submanifold S, the Poincare constant is bounded below by the first nonzero eigenvalue of the Laplace-Beltrami operator on S, for low temperature.