用签名距离提升路径数据的局部回归效率与稳定性
Local regression on path spaces with signature metrics
- 结合签名变换与局部核回归，实现路径数据的高效建模
- 在有限样本下证明签名距离优于传统度量，具良好统计性质
- 对异常值鲁棒，适合时间序列分类与随机微分方程学习
我们研究路径型数据的非参数回归与分类问题。提出一种基于粗糙路径理论中签名变换的函数型Nadaraya-Watson估计器，利用迭代积分编码序列数据，通过签名诱导的距离在自然度量空间中直接比较路径。该方法将签名距离融入经典核回归框架，在避免大规模核矩阵计算瓶颈的同时保持计算高效。我们建立了有限样本收敛边界，表明签名距离在无限维设置下优于传统度量。进一步提出对抗异常值的鲁棒签名变体，提升实际性能。在合成数据与真实数据（包括随机微分方程学习和时间序列分类）上的应用显示，该方法在精度上具有竞争力，且计算优势显著。
原文摘要 · Abstract (English)
We study nonparametric regression and classification for path-valued data. We introduce a functional Nadaraya-Watson estimator that combines the signature transform from rough path theory with local kernel regression. The signature transform provides a principled way to encode sequential data through iterated integrals, enabling direct comparison of paths in a natural metric space. Our approach leverages signature-induced distances within the classical kernel regression framework, achieving computational efficiency while avoiding the scalability bottlenecks of large-scale kernel matrix operations. We establish finite-sample convergence bounds demonstrating favorable statistical properties of signature-based distances compared to traditional metrics in infinite-dimensional settings. We propose robust signature variants that provide stability against outliers, enhancing practical performance. Applications to both synthetic and real-world data - including stochastic differential equation learning and time series classification - demonstrate competitive accuracy while offering significant computational advantages over existing methods.
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