Log-PDE Methods for Rough Signature Kernels
This addresses a computational bottleneck in machine learning for multivariate time series analysis, though it appears incremental as an extension from bounded variation to rough paths.
The paper tackles the computational intractability of signature kernels for highly oscillatory rough paths by developing a novel system of PDEs that uses higher-order iterated integrals as coefficients, achieving higher-order approximation with quantitative error bounds.
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.