Log-PDE Methods for Rough Signature Kernels

TL;DR

This paper introduces Log-PDE methods for approximating rough signature kernels in multivariate time series analysis, providing higher order accuracy and computational efficiency for highly oscillatory paths.

Contribution

It extends PDE-based signature kernel computation to rough paths using a novel system of PDEs with higher order iterated integrals, improving accuracy and efficiency.

Findings

Unique solution for the PDE system is established.

Quantitative error bounds demonstrate higher order approximation.

Method is effective for highly oscillatory input paths.

Abstract

Signature kernels, inner products of path signatures, underpin several machine learning algorithms for multivariate time series analysis. For bounded variation paths, signature kernels were recently shown to solve a Goursat PDE. However, existing PDE solvers only use increments as input data, leading to first order approximation errors. These approaches become computationally intractable for highly oscillatory input paths, as they have to be resolved at a fine enough scale to accurately recover their signature kernel, resulting in significant time and memory complexities. In this paper, we extend the analysis to rough paths, and show, leveraging the framework of smooth rough paths, that the resulting rough signature kernels can be approximated by a novel system of PDEs whose coefficients involve higher order iterated integrals of the input rough paths. We show that this system of PDEs admits a unique solution and establish quantitative error bounds yielding a higher order approximation to rough signature kernels.