Fourier computation of empirical diffusion metrics

Abstract

We give a finite Fourier algorithm for statistical inference on inverse weighted-Laplacian metric entries estimated by empirical heat-kernel Sinkhorn normalization. For every fixed positive overhead exponent, the algorithm uses a deterministic number of exact-real operations bounded by the sample size to the power one plus that exponent. It preserves the population-centered Gaussian limit, consistent joint influence covariance, and the ordinary nonparametric bootstrap of the exact metric estimator. The density is fixed, smooth and strictly positive on a fixed-dimensional flat torus, and the number of prescribed smooth scores is fixed. The proof establishes an empirical spectral gap proportional to bandwidth, a logarithmic number of actual half-scaling iterations, and complete original-law error bounds for the inverse response and all covariance products. Positive normalized Fourier features and reduced Woodbury solves pay the full feature Gram and factorization costs. Higher fixed-order bias extrapolation permits the bandwidth needed for each prescribed operation bound.