Residual Stopping and Pointwise Inference for Nonuniform Entropic Self-Transport

Abstract

We study observable residual stopping for the half-self-Sinkhorn iteration with iid samples from a fixed positive smooth density on the one- or two-dimensional torus. For the normalized periodic Gaussian, assume that ε→0, nε^(3d/2)→∞, and nε^(d/2+3)→0. Starting from the constant scaling, we stop at the first projective square-root-mass residual below t_n=n^(-1/2)ε^(-γ), where 0<γ<d/4 is fixed. The stopped score has the same pointwise Gaussian limit, centered at the true density score, as the exactly scaled empirical estimator. Its attained residual lies between ct_n and t_n with probability tending to one for every fixed 0<c<1/2, although √(n) t_n→∞. The stopping index is asymptotic to (d/4-γ)log(1/ε)/log2 in probability. For the coefficient-one threshold, uniform sampling already gives a different score limit when γ≥ d/4. The proof retains the iteration-dependent empirical-norm error and uses a normalized-row cancellation to control nonuniform initialization in dimension two.