Numerical schemes for signature kernels
File(s)
Accepted version
Supporting information
Author(s)
Cass, Thomas
Piatti, Francesco
Pei, Jeffrey
Type
Journal Article
Abstract
Signature kernels have become a powerful tool in kernel methods for sequential data. In “The Signature Kernel is the solution of a Goursat PDE” [35], the authors introduced a kernel trick showing that, for continuously differentiable paths, the signature kernel satisfies a hyperbolic PDE of Goursat type in two independent time variables. While finite difference methods have been explored for this PDE, they suffer from accuracy and stability issues when handling highly oscillatory inputs. In this work, we propose two advanced numerical schemes that approximate the solution using polynomial representations of boundary conditions and employing either approximation or interpolation techniques. We prove the convergence of the polynomial approximation scheme and demonstrate experimentally that both methods achieve several orders of magnitude improvement in mean absolute percentage error (MAPE) over finite difference schemes without increasing computational complexity. These algorithms are implemented in a publicly available Python library: https://github.com/FrancescoPiatti/polysigkernel.
Date Issued
2025-12-01
Date Acceptance
2025-08-21
Citation
SIAM Journal on Numerical Analysis, 2025, 63 (6), pp.2371-2394
ISSN
0036-1429
Publisher
Society for Industrial and Applied Mathematics
Start Page
2371
End Page
2394
Journal / Book Title
SIAM Journal on Numerical Analysis
Volume
63
Issue
6
Copyright Statement
Copyright © 2025 Society for Industrial and Applied Mathematics. This is the author’s accepted manuscript made available under a CC-BY licence in accordance with Imperial’s Research Publications Open Access policy (www.imperial.ac.uk/oa-policy)
License URL
Publication Status
Published
Date Publish Online
2025-12-11