Signature asymptotics, empirical processes, and optimal transport
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路径签名渐近理论、经验过程与最优传输
- 研究问题
- 路径签名的渐近行为如何与经验过程及 Wasserstein 距离相联系，并能否借此降低既有极限结果对路径光滑性的要求？
- 方法与思路
- 作者将 Hambly–Lyons 极限重新解释为同一分布下两组独立样本经验测度之间的 Wasserstein 距离渐近行为，并使用已有收敛率结果。
- 材料陈述的贡献
- 摘要称，该联系把原有结论从三阶连续可微曲线推广至一大类二阶连续可微曲线，并给出通过二阶微分方程计算极限的方式。
- 阅读关注点
- 应核对路径几何如何决定采样分布、经验测度与签名之间的对应关系，以及二阶可微曲线仍需满足哪些额外条件。
- 阅读边界
- 这是 arXiv 预印本，同行评审及期刊接收未核实；摘要未说明曲线类别的完整限制、微分方程边界条件或统计应用效果，不能推广到任意路径。
原文摘要
Rough path theory provides one with the notion of signature, a graded family of tensors which characterise, up to a negligible equivalence class, and ordered stream of vector-valued data. In the last few years, use of the signature has gained traction in time-series analysis, machine learning , deep learning and more recently in kernel methods. In this article, we lay down the theoretical foundations for a connection between signature asymptotics, the theory of empirical processes, and Wasserstein distances, opening up the landscape and toolkit of the second and third in the study of the first. Our main contribution is to show that the Hambly-Lyons limit can be reinterpreted as a statement about the asymptotic behaviour of Wasserstein distances between two independent empirical measures of samples from the same underlying distribution. In the setting studied here, these measures are derived from samples from a probability distribution which is determined by geometrical properties of the underlying path. The general question of rates of convergence for these objects has been studied in depth in the recent monograph of Bobkov and Ledoux. By using these results, we generalise the original result of Hambly and Lyons from $C^3$ curves to a broad class of $C^2$ ones. We conclude by providing an explicit way to compute the limit in terms of a second-order differential equation.