The Signature Kernel is the solution of a Goursat PDE

TL;DR

This paper reveals that the signature kernel for sequential data solves a Goursat PDE, enabling efficient computation and GPU parallelization, with extensions to rough paths and empirical validation in machine learning tasks.

Contribution

It establishes a novel connection between the signature kernel and Goursat PDEs, providing an efficient PDE-based kernel computation method and extending analysis to rough paths.

Findings

PDE-based signature kernel can be computed efficiently using hyperbolic PDE solvers.

The method is well-suited for GPU parallelization, reducing computational complexity.

Empirical results demonstrate the kernel's effectiveness in machine learning applications.

Abstract

Recently, there has been an increased interest in the development of kernel methods for learning with sequential data. The signature kernel is a learning tool with potential to handle irregularly sampled, multivariate time series. In "Kernels for sequentially ordered data" the authors introduced a kernel trick for the truncated version of this kernel avoiding the exponential complexity that would have been involved in a direct computation. Here we show that for continuously differentiable paths, the signature kernel solves a hyperbolic PDE and recognize the connection with a well known class of differential equations known in the literature as Goursat problems. This Goursat PDE only depends on the increments of the input sequences, does not require the explicit computation of signatures and can be solved efficiently using state-of-the-arthyperbolic PDE numerical solvers, giving a kernel trick for the untruncated signature kernel, with the same raw complexity as the method from "Kernels for sequentially ordered data", but with the advantage that the PDE numerical scheme is well suited for GPU parallelization, which effectively reduces the complexity by a full order of magnitude in the length of the input sequences. In addition, we extend the previous analysis to the space of geometric rough paths and establish, using classical results from rough path theory, that the rough version of the signature kernel solves a rough integral equation analogous to the aforementioned Goursat PDE. Finally, we empirically demonstrate the effectiveness of our PDE kernel as a machine learning tool in various machine learning applications dealing with sequential data. We release the library sigkernel publicly available at https://github.com/crispitagorico/sigkernel.