Sharp Poincaré Interpolation Along Wasserstein Geodesics

What the math gives to ML

The paper provides an endpoint-to-interpolation principle: strong log-concavity at two endpoint distributions yields a sharp Poincare bound for every intermediate Wasserstein distribution, without requiring curvature control of the intermediate density. The transferable asset is the linear interpolation of the Poincare scale, rather than the Poincare constant itself, together with a guarantee for arbitrary test functions. A practical neural-network use is to train on coupled Wasserstein interpolants and penalize violations of the resulting variance-versus-input-Jacobian inequality, producing a path-wise stability regularizer whose coefficient is determined only from endpoint concentration estimates.