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Local regression on path spaces with signature metrics

Christian Bayer, Davit Gogolashvili, Luca Pelizzari

TL;DR

The paper develops a local regression framework for path-valued data by embedding paths via the signature transform and using a signature-induced semi-metric within a Nadaraya–Watson estimator. It provides finite-sample convergence guarantees, with error rates governed by small-ball probabilities; the authors show that truncating the signature yields Euclidean-like rates with effective dimension $\nu(N)$, while the full signature space can incur slower, exponential-type rates. They extend the theory to rough differential equations, where the Itô–Lyons map is Hölder continuous in signature distance, and address robustness through a robust signature variant. Empirically, the method improves learning of SDE solution maps and achieves competitive time-series classification results with favorable computational efficiency compared to kernel-based methods and DTW baselines. The framework offers a unified, scalable approach for nonparametric regression and classification on infinite-dimensional path spaces with solid theoretical guarantees and practical impact for sequential data analysis.

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.

Local regression on path spaces with signature metrics

TL;DR

The paper develops a local regression framework for path-valued data by embedding paths via the signature transform and using a signature-induced semi-metric within a Nadaraya–Watson estimator. It provides finite-sample convergence guarantees, with error rates governed by small-ball probabilities; the authors show that truncating the signature yields Euclidean-like rates with effective dimension , while the full signature space can incur slower, exponential-type rates. They extend the theory to rough differential equations, where the Itô–Lyons map is Hölder continuous in signature distance, and address robustness through a robust signature variant. Empirically, the method improves learning of SDE solution maps and achieves competitive time-series classification results with favorable computational efficiency compared to kernel-based methods and DTW baselines. The framework offers a unified, scalable approach for nonparametric regression and classification on infinite-dimensional path spaces with solid theoretical guarantees and practical impact for sequential data analysis.

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.
Paper Structure (20 sections, 12 theorems, 116 equations, 1 figure, 2 tables)

This paper contains 20 sections, 12 theorems, 116 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Notations and Background
  4. Local regression with signatures
  5. Convergence analysis
  6. Signature semi-metrics on path-spaces
  7. Solution maps of rough differential equations
  8. Robustification
  9. Application
  10. Learning the solution map of SDEs
  11. Time Series Classification
  12. Conclusion
  13. Convergence Guarantee
  14. Bound on $B$.
  15. Bound on $A_1$.
  16. ...and 5 more sections

Key Result

Theorem 4

Let $Y \in [-R,R]$ and let $F \in \mathcal{F}_{\beta}$ with smoothness parameter $\beta \in (0,1]$. Consider the estimator $\widehat{F}$ defined in eq:NW_estimator_regression, and assume that the kernel $K$ is compactly supported and satisfies for some constants $0 < b \leq B < \infty$. For any $\delta \in (0,1)$ and $M$ satisfying with probability at least $1-\delta$ the following bound holds

Figures (1)

  • Figure 1: Scatter plots of the testing data $\{(Y^{(m)}, \widehat{Y}^{(m)}):m \in I_{te} \}$, using signature and supremum metrics in \ref{['eq:estimator_SDE']}. At truncation level $N=4$, the point ${\color{orange} \bullet}$ near $(1.2,0)$ illustrates an outlier of the basic method Sig; see the discussion in Section \ref{['sec:robustification']}.

Theorems & Definitions (28)

  • Definition 1
  • Remark 2
  • Definition 3
  • Theorem 4
  • Remark 5
  • Remark 6
  • Lemma 7
  • Example 1
  • Proposition 9
  • Remark 10
  • ...and 18 more