非参数薛定谔桥时间序列生成器：算法、收敛性分析及应用
Nonparametric Schrödinger Bridge Time Series Generator: Algorithm, Convergence Analysis and Applications
AI总结：
该研究对SBTS生成器进行收敛性分析，证明其在参数合适时收敛，还验证了其替换参考测度的灵活性与框架鲁棒性，为时间序列生成提供了理论与实证支撑。
AI中文摘要：
我们对薛定谔桥时间序列（Schrödinger Bridge Time Series，SBTS）数据生成器进行收敛性分析。从正则化形式出发，该形式中数据集合以规定概率与标准多元高斯分布混合，我们证明：当集合规模和核带宽选择恰当时，欧拉-丸山（Euler-Maruyama）离散化以半阶收敛速度收敛至混合目标分布。我们进一步证明，当混合概率趋于零时，正则化分布收敛至原始目标分布。该分析同时考虑了集合近似误差、核近似误差和时间离散化误差，因此为SBTS生成器提供了完整的分布收敛结果。在实证研究中，我们通过将维纳参考测度替换为更一般随机微分方程（SDE）诱导的路径测度，进一步检验了该方法的灵活性。数值实验表明，基于原始维纳参考测度和SDE诱导参考测度的方案均取得了可比性能，证明了SBTS框架的鲁棒性与稳定性。
英文摘要：
We conduct a convergence analysis for the Schrödinger Bridge Time Series (SBTS) data generator. Starting from a regularized formulation in which the data ensemble is mixed with a standard multivariate Gaussian distribution with a prescribed probability, we prove that the Euler-Maruyama discretization converges to the mixed target distribution with half-order convergence rate, provided that the ensemble size and kernel bandwidth are chosen appropriately. We further show that the regularized distribution converges to the original target distribution as the mixing probability tends to zero. The analysis simultaneously accounts for the ensemble approximation error, kernel approximation error, and time-discretization error, and therefore provides a full distributional convergence result for the SBTS generator. Empirically, we further examine the flexibility of the method by replacing the Wiener reference measure with the path measure induced by a more general SDE. The numerical experiments show that the schemes based on both the original Wiener reference measure and the SDE-induced reference measure achieve comparable performance, demonstrating the robustness and stability of the SBTS framework.
