Local Regression on Path Spaces with Signature Metrics

Christian Bayer, Davit Gogolashvili, Luca Pelizzari
Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, PMLR 300:2755-2763, 2026.

Abstract

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v300-bayer26a, title = { Local Regression on Path Spaces with Signature Metrics }, author = {Bayer, Christian and Gogolashvili, Davit and Pelizzari, Luca}, booktitle = {Proceedings of The 29th International Conference on Artificial Intelligence and Statistics}, pages = {2755--2763}, year = {2026}, editor = {Khan, Emtiyaz and Li, Yingzhen and Solin, Arno and Ramdas, Aaditya}, volume = {300}, series = {Proceedings of Machine Learning Research}, month = {02--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v300/main/assets/bayer26a/bayer26a.pdf}, url = {https://proceedings.mlr.press/v300/bayer26a.html}, abstract = { 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. } }
Endnote
%0 Conference Paper %T Local Regression on Path Spaces with Signature Metrics %A Christian Bayer %A Davit Gogolashvili %A Luca Pelizzari %B Proceedings of The 29th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2026 %E Emtiyaz Khan %E Yingzhen Li %E Arno Solin %E Aaditya Ramdas %F pmlr-v300-bayer26a %I PMLR %P 2755--2763 %U https://proceedings.mlr.press/v300/bayer26a.html %V 300 %X 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.
APA
Bayer, C., Gogolashvili, D. & Pelizzari, L.. (2026). Local Regression on Path Spaces with Signature Metrics . Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 300:2755-2763 Available from https://proceedings.mlr.press/v300/bayer26a.html.

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