Wasserstein Propagation for Reverse Diffusion under Weak Log-Concavity: Exploiting Metric Mismatch via One-Switch Routing

arXiv:2603.19670v2 Announce Type: replace Abstract: Existing analyses of reverse diffusion typically propagate sampling error in the Euclidean geometry underlying \(\Wtwo\) throughout the reverse trajectory. Under weak log-concavity, this can be suboptimal: Gaussian smoothing may create contraction first at large separations, while short-scale Euclidean dissipativity is still absent. We show that exploiting this metric mismatch can yield strictly sharper end-to-end \(\Wtwo\) bounds than direct full-horizon Euclidean propagation on mismatch windows. Our analysis derives an explicit radial lower profile for the learned reverse drift, whose far-field and near-field limits quantify the contraction reserve and the residual Euclidean load, respectively. This profile determines admissible switch times and leads to a one-switch routing theorem: reflection coupling damps initialization mismatch, pre-switch score forcing, and pre-switch discretization in an adapted concave transport metric; a single \(p\)-moment interpolation converts the damped switch-time discrepancy back to \(\Wtwo\); and synchronous coupling propagates the remaining error over the late Euclidean window. Under \(L^2\) score-error control, a one-sided monotonicity condition on the score error, and standard well-posedness and coupling assumptions, we obtain explicit non-asymptotic end-to-end \(\Wtwo\) guarantees, a scalar switch-selection objective, and a conversion exponent \(\theta_p=(p-2)/(2(p-1))\) that cannot be improved uniformly within the affine-tail concave class under the same \(p\)-moment switch assumption. For a fixed switch, the routed and direct Euclidean bounds share the same late-window term, so any strict improvement is entirely an early-window effect.