Optimal Riemannian metric for Poincaré inequalities and how to ideally precondition Langevin dynamics

TL;DR

The paper develops a Riemannian generalization of the Poincaré inequality by introducing a space-dependent metric W to minimize the Poincaré constant and improve Langevin dynamics convergence. Under the moment-measure assumption, an optimal metric exists with C(μ,W)=1 and W forming a Stein kernel, connecting functional inequalities to Stein's method. The authors recast the problem as a concave spectral optimization, and propose a gradient-based finite element algorithm to compute W efficiently, yielding metrics that reveal the geometry of the target measure and serve as effective preconditioners for Langevin sampling. Numerical experiments in 2D on multiple measures illustrate how the optimal metric concentrates anisotropy across modes, accelerates mixing, and enables robust performance of preconditioned Langevin dynamics. The work provides both theoretical and computational tools to design geometry-informed sampling schemes with potential impact in MCMC and stochastic optimization.

Abstract

Poincaré inequality is a fundamental property that rises naturally in different branches of mathematics. The associated Poincaré constant plays a central role in many applications since it governs the convergence of various practical algorithms. For instance, the convergence rate of the Langevin dynamics is exactly given by the Poincaré constant. This paper investigates a Riemannian version of Poincaré inequality where a positive definite weighting matrix field (\emph{i.e.} a Riemannian metric) is introduced to improve the Poincaré constant, and therefore the performances of the associated algorithm. Assuming the underlying measure is a \emph{moment measure}, we show that an optimal metric exists and the resulting Poincaré constant is 1. We demonstrate that such optimal metric is necessarily a \emph{Stein kernel}, offering a novel perspective on these complex but central mathematical objects that are hard to obtain in practice. We further discuss how to numerically obtain the optimal metric by deriving an implementable optimization algorithm. The resulting method is illustrated in a few simple but nontrivial examples, where solutions are revealed to be rather sophisticated. We also demonstrate how to design efficient Langevin-based sampling schemes by utilizing the precomputed optimal metric as a preconditioner.

Figures (6)

• Figure 1: The four benchmark measures and the finite element discretization of their support (cyan indicates low probability density and purple indicates high probability density). The number of elements in the mesh are respectively $M_1=5718$, $M_2=10256$, $M_3=4610$ and $M_4=5742$.
• Figure 2: (Tri-modal $\mu_1$) Evolution of the five first nonzero eigenvalues of the diffusion operator $\mathcal{L}_\mu^{W}$ during the iterative process of the gradient-ascent methods, with $\rho=0.01$ and $\alpha=0$ (Gradient ascent), $\alpha=0.5$ (Momentum) and $\alpha_k = 1-3/(5+k)$ (Nesterov).
• Figure 3: (Tri-modal $\mu_1$) Eigenvectors $u_2^W,\hdots,u_6^W$ of the diffusion operator $\mathcal{L}_{\mu_1}^{W}$ for the constant metric $W^{(0)}=I_d \frac{\mathop{\mathrm{tr}}\nolimits(\mathop{\mathrm{Cov}}\nolimits_\mu)}{d}$ (top row) and the computed metric $W^{(k)}$ obtained using $k=100$ iterations of Nesterov's acceleration.
• Figure 4: Metric $W^{(k)}$ computed with $k=100$ iterations of Nesterov's acceleration. The colorbar corresponds to $\log_{10}(\mathop{\mathrm{tr}}\nolimits(W^{(k)}))$, which is the amplitude of $W^{(k)}$ in log-scale. The ellipses represent the local anisotropy of $W^{(k)}$.
• Figure 5: Drift $(\mathrm{div}(W)- W\nabla V_i)$ for the four considered measures $\mathrm{d}\mu_i\propto\exp(-V_i)\mathrm{d} x$, $i=1,\hdots,4$, with either the constant metric $W\propto I_d$ (Figures \ref{['fig:drift_a']} and \ref{['fig:drift_d']}) or the optimal metric $W$.

Theorems & Definitions (19)

• Proposition 1
• Theorem 2.1: Existence of a solution to \ref{['eq:minC']}
• proof
• Proposition 2
• proof
• Example 1
• Corollary 1
• proof
• Theorem 3.1
• proof