# Local regression on path spaces with signature metrics

arXiv: https://arxiv.org/abs/2510.16728 · HTML: https://mlsanity.com/paper/2510.16728/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 62 references that are in the map.
- First posted on arXiv 19 Oct 2025.

## Builds on (most cited first, 23 of 23)

- [The UEA multivariate time series classification archive, 2018](https://mlsanity.com/paper/1811.00075/) (2018)
- [Kernels for sequentially ordered data](https://mlsanity.com/paper/1601.08169/) (2016)
- [Deep Signature Transforms](https://mlsanity.com/paper/1905.08494/) (2019)
- [The Signature Kernel is the solution of a Goursat PDE](https://mlsanity.com/paper/2006.14794/) (2020)
- [Signature moments to characterize laws of stochastic processes](https://mlsanity.com/paper/1810.10971/) (2018)
- [The iisignature library: efficient calculation of iterated-integral signatures and log signatures](https://mlsanity.com/paper/1802.08252/) (2018)
- [A review on distance based time series classification](https://mlsanity.com/paper/1806.04509/) (2018)
- [Signature Methods in Machine Learning](https://mlsanity.com/paper/2206.14674/) (2022)
- [Embedding and learning with signatures](https://mlsanity.com/paper/1911.13211/) (2019)
- [SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data](https://mlsanity.com/paper/2105.04211/) (2021)
- [Developing the Path Signature Methodology and its Application to Landmark-based Human Action Recognition](https://mlsanity.com/paper/1707.03993/) (2017)
- [Rough Transformers: Lightweight and Continuous Time Series Modelling through Signature Patching](https://mlsanity.com/paper/2405.20799/) (2024)
- [Lecture notes on rough paths and applications to machine learning](https://mlsanity.com/paper/2404.06583/) (2024)
- [Random Fourier Signature Features](https://mlsanity.com/paper/2311.12214/) (2023)
- [The Signature Kernel](https://mlsanity.com/paper/2305.04625/) (2023)
- [Functional linear regression with truncated signatures](https://mlsanity.com/paper/2006.08442/) (2020)
- [Nowcasting using regression on signatures](https://mlsanity.com/paper/2305.10256/) (2023)
- [On Consistency of Signature Using Lasso](https://mlsanity.com/paper/2305.10413/) (2023)
- [Learning the Dynamics of Sparsely Observed Interacting Systems](https://mlsanity.com/paper/2301.11647/) (2023)
- [Nonparametric Regression and Classification with Functional, Categorical, and Mixed Covariates](https://mlsanity.com/paper/2111.03115/) (2021)
- [Variable selection in functional regression models: a review](https://mlsanity.com/paper/2401.14867/) (2024)
- [Bayesian bandwidth estimation for a nonparametric functional regression model with mixed types of regressors and unknown error density](https://mlsanity.com/paper/1403.1913/) (2014)
- [Expected Signature Kernels for Rough Paths](https://mlsanity.com/paper/2509.07893/) (2025)

## Built on by (most cited first, 1 of 1)

- [Branched Signature Kernel Solvers for ODEs with rough Single-Trajectory signals](https://mlsanity.com/paper/2605.25826/) (2026)

## Abstract (arXiv)

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.