Direct Estimation of Schr\"odinger Bridge Time-Series Drifts: Finite-Sample, Asymptotic, and Adaptive Guarantees

TL;DR

This paper introduces a direct kernel-based estimator for Schr"odinger bridge drifts from i.i.d. data, providing finite-sample, asymptotic, and adaptive theoretical guarantees, and demonstrating its effectiveness through synthetic experiments.

Contribution

It presents a novel direct estimation method for Schr"odinger bridge drifts that isolates statistical error and offers minimax-rate optimal adaptive bandwidth selection with theoretical guarantees.

Findings

Proves a uniform non-asymptotic bound for bandwidths

Establishes a pointwise CLT under undersmoothing

Shows the adaptive bandwidth selector is minimax-rate optimal

Abstract

We study nonparametric estimation of Schr\"odinger bridge (SB) drifts from i.i.d.\ data observed on a single time interval. Starting from the conditional-ratio form of the Schr\"odinger bridge time-series (SBTS) drift formula, we analyze a direct Nadaraya--Watson plug-in estimator built from kernelized numerator and denominator terms. Unlike recent SB analyses based on entropic-OT potentials, Sinkhorn iterations, or iterative bridge solvers, our approach works directly at the drift level and isolates \emph{statistical error} from optimization, approximation, and discretization error. Under H\"older regularity, a marginal-density floor, and bounded support, we prove a uniform non-asymptotic bound for admissible bandwidth pairs, a pointwise CLT under genuine undersmoothing, and an adaptive bandwidth selector satisfying an oracle inequality. We also prove a pivot-local minimax lower bound which, through an explicit uniform pivot, yields a global minimax lower bound under transparent compatibility conditions; hence the adaptive selector is minimax-rate optimal up to logarithmic factors. Synthetic experiments provide theorem-targeted diagnostics for finite-sample scaling, Gaussian approximation, and adaptive behavior.