Numerical Schemes for Signature Kernels

TL;DR

This paper introduces advanced polynomial-based numerical schemes for solving the PDE associated with signature kernels, significantly improving accuracy and scalability over traditional finite difference methods, with practical GPU implementation.

Contribution

The authors develop and theoretically validate two polynomial approximation schemes for the signature kernel PDE, enhancing accuracy and computational efficiency.

Findings

Achieved several orders of magnitude reduction in MAPE compared to finite difference methods.

Algorithms can be GPU-parallelized, reducing complexity from quadratic to linear.

Implemented a publicly available Python library for the proposed methods.

Abstract

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.