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Scalable Signature Kernel Computations for Long Time Series via Local Neumann Series Expansions

by M. Tamayo-Rios and A. Schell and R. Alaifari

(Report number 2025-09)

Abstract
The signature kernel is a recent state-of-the-art tool for analyzing high-dimensional sequential data, valued for its theoretical guarantees and strong empirical performance. In this paper, we present a novel method for efficiently computing the signature kernel of long, high-dimensional time series via dynamically truncated recursive local power series expansions. Building on the characterization of the signature kernel as the solution of a Goursat PDE, our approach employs tilewise Neumann-series expansions to derive rapidly converging power series approximations of the signature kernel that are locally defined on subdomains and propagated iteratively across the entire domain of the Goursat solution by exploiting the geometry of the time series. Algorithmically, this involves solving a system of interdependent local Goursat PDEs by recursively propagating boundary conditions along a directed graph via topological ordering, with dynamic truncation adaptively terminating each local power series expansion when coefficients fall below machine precision, striking an effective balance between computational cost and accuracy. This method achieves substantial performance improvements over state-of-the-art approaches for computing the signature kernel, providing (a) adjustable and superior accuracy, even for time series with very high roughness; (b) drastically reduced memory requirements; and (c) scalability to efficiently handle very long time series (e.g., with up to half a million points or more) on a single GPU. These advantages make our method particularly well-suited for rough-path-assisted machine learning, financial modeling, and signal processing applications that involve very long and highly volatile data.

Keywords: signature kernel, Goursat, PDE, GPU, time series analysis

@Techreport{TSA25_1130,
  author = {M. Tamayo-Rios and A. Schell and R. Alaifari},
  title = {Scalable Signature Kernel Computations for Long Time Series via Local Neumann Series Expansions},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2025-09},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2025/2025-09.pdf },
  year = {2025}
}

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