Leading bias and positive correction of heat-kernel Sinkhorn metrics

Overview

Abstract

We identify the variable-density leading bias of the exact population heat-kernel Sinkhorn tensor on a flat torus. After ground-state conjugation, its second-order quadratic-form coefficient is minus one forty-eighth of the Wasserstein Hessian of Fisher information. The remainder is fourth order from H^6 to L^2, uniformly in every fixed parameter-derivative order along smooth positive density families. An explicit nonconstant density has both positive and negative bias directions. A compact selfadjoint density-dependent correction removes this bias, remains strictly positive on the entire tangent complement, and preserves the original completed tangent form space and square-root range with equivalent norms. The compact correction and ordinary two-bandwidth Richardson extrapolation have sharp H^r→ L^2 error h^(1+r/2) for 0≤ r≤6. Full L^2 accuracy is first order. Uniform quadratic-form comparison with the original tensor supplies positivity in its natural topology, rather than a uniform positive L^2 lower bound. The proof retains moving-nullspace and relative elliptic parameter estimates and uses nonnegative heat times. An exactly normalized nonconstant two-parameter family gives a certified corrected-metric computation with separate continuum residual, differential-form and quadrature errors. For prescribed analytic density families sufficiently close to uniform, we also prove normalization-inclusive fourth-order finite metric approximation in every fixed parameter-derivative order. In fixed dimension d, per-coordinate Fourier resolution and iteration counts are logarithmic in inverse bandwidth; the numbers of modes and quadrature nodes grow as the dth power of that logarithm.

Keywords Sinkhorn metrics · heat kernel · leading bias · Fisher information · positive correction · analytic densities · certified approximation