直接估计量子桥时间序列漂移，兼具理论保障与自适应能力。
Direct Estimation of Schrödinger Bridge Time-Series Drifts: Finite-Sample, Asymptotic, and Adaptive Guarantees
- 基于核方法直接构建漂移估计器，分离统计误差与优化误差。
- 在有限样本下给出统一上界，点态收敛满足中心极限定理。
- 自适应带宽选择达到极小极大最优率，适合高维数据建模。
我们研究从独立同分布数据中非参数估计单个时间区间上的薛定谔桥（SB）漂移。从薛定谔桥时间序列（SBTS）漂移公式的条件比值形式出发，分析一种直接的Nadaraya-Watson插值估计器，该估计器由核化的分子和分母项构成。不同于近期基于熵正则化最优传输势、Sinkhorn迭代或迭代桥求解器的分析方法，我们的方法直接作用于漂移层面，并将统计误差与优化、近似及离散化误差分离。在霍尔德正则性、边缘密度下界和有界支撑条件下，我们证明了可接受带宽对的统一非渐近上界，以及真实欠平滑下的点态中心极限定理；并构造了一个满足奥拉克不等式的自适应带宽选择器。此外，我们还证明了枢轴局部极小极大下界，通过显式统一枢轴得到在透明相容条件下的全局极小极大下界；因此，自适应选择器在对数因子内达到极小极大最优率。合成实验提供了针对有限样本尺度、高斯逼近和自适应行为的定理目标诊断。
原文摘要 · Abstract (English)
We study nonparametric estimation of Schrödinger bridge (SB) drifts from i.i.d.\ data observed on a single time interval. Starting from the conditional-ratio form of the Schrödinger 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 Hölder 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.
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