# Numerical Schemes for Signature Kernels

arXiv: https://arxiv.org/abs/2502.08470 · HTML: https://mlsanity.com/paper/2502.08470/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 36 references that are in the map.
- First posted on arXiv 12 Feb 2025.

## Builds on (most cited first, 11 of 11)

- [Kernels for sequentially ordered data](https://mlsanity.com/paper/1601.08169/) (2016)
- [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)
- [Distribution Regression for Sequential Data](https://mlsanity.com/paper/2006.05805/) (2020)
- [Neural signature kernels as infinite-width-depth-limits of controlled ResNets](https://mlsanity.com/paper/2303.17671/) (2023)
- [Framing RNN as a kernel method: A neural ODE approach](https://mlsanity.com/paper/2106.01202/) (2021)
- [SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data](https://mlsanity.com/paper/2105.04211/) (2021)
- [SK-Tree: a systematic malware detection algorithm on streaming trees via the signature kernel](https://mlsanity.com/paper/2102.07904/) (2021)
- [Random Fourier Signature Features](https://mlsanity.com/paper/2311.12214/) (2023)
- [Kernel Approximation Methods for Speech Recognition](https://mlsanity.com/paper/1701.03577/) (2017)
- [Transportation Marketplace Rate Forecast Using Signature Transform](https://mlsanity.com/paper/2401.04857/) (2024)

## Built on by (most cited first, 3 of 3)

- [Scalable Signature Kernel Computations for Long Time Series via Local Neumann Series Expansions](https://mlsanity.com/paper/2502.20392/) (2025)
- [Random Controlled Differential Equations](https://mlsanity.com/paper/2512.23670/) (2025)
- [Learning Manifold and Ito Dynamics with Branched Neural Rough Differential Equations](https://mlsanity.com/paper/2606.05272/) (2026)

## Abstract (arXiv)

Signature kernels have emerged as a powerful tool within kernel methods for sequential data. In the paper "The Signature Kernel is the solution of a Goursat PDE", the authors identify a kernel trick that demonstrates that, for continuously differentiable paths, the signature kernel satisfies a Goursat problem for a hyperbolic partial differential equation (PDE) in two independent time variables. While finite difference methods have been explored for this PDE, they face limitations in accuracy and stability when handling highly oscillatory inputs. In this work, we introduce two advanced numerical schemes that leverage polynomial representations of boundary conditions through either approximation or interpolation techniques, and rigorously establish the theoretical convergence of the polynomial approximation scheme. Experimental evaluations reveal that our approaches yield improvements of several orders of magnitude in mean absolute percentage error (MAPE) compared to traditional finite difference schemes, without increasing computational complexity. Furthermore, like finite difference methods, our algorithms can be GPU-parallelized to reduce computational complexity from quadratic to linear in the length of the input sequences, thereby improving scalability for high-frequency data. We have implemented these algorithms in a dedicated Python library, which is publicly available at: https://github.com/FrancescoPiatti/polysigkernel.