Leading bias and positive correction of heat-kernel Sinkhorn metrics

Abstract

We identify the leading bias of the population heat-kernel Sinkhorn tensor at smooth strictly positive densities on a flat torus. Its second-order quadratic-form coefficient is minus one forty-eighth of the Wasserstein Hessian of Fisher information, with a fourth-order H⁶ → L² remainder uniform under every fixed order of density-parameter differentiation. An explicit density has bias directions of both signs. A compact same-bandwidth correction removes the bias, remains positive on the full tangent complement, and preserves the original completed form space and square-root range with equivalent norms. Its sharp Hʳ → L² rates are h^(1+r/2) for 0 ≤ r ≤ 6; full L² accuracy remains first order. On immersed finite-dimensional density families, corrected metrics and intrinsic curvature have fourth-order accuracy, with relative fourth-order comparison of path distances restricted to a fixed compact parameter region. Geodesic initial-value solutions and parallel transport have the same rate on fixed time intervals within a compact parameter chart. Computational appendices give certified metric values and normalization-inclusive approximation for prescribed analytic families close to uniform, with logarithmic per-coordinate resolution under explicit coefficient and tail inputs.