Log-PDE Methods for Rough Signature Kernels

2024-04-01

AI Summary

Computing the signature kernel for highly oscillatory multivariate time series is hindered by existing methods that rely on path increments, introducing first-order approximation errors and demanding fine discretization—leading to high computational and memory complexity. Method: We extend the Goursat-type partial differential equation (PDE) framework to rough paths, constructing a novel PDE system whose coefficients are higher-order iterated integrals. Contribution/Results: This work establishes, for the first time, a well-posed higher-order PDE model for the rough signature kernel, circumventing the accuracy limitations of increment-based approaches. We prove existence and uniqueness of the solution and derive an explicit quantitative error bound. The resulting algorithm achieves higher-than-first-order kernel approximation while maintaining low time and space complexity. Our method is particularly suited for modeling strongly oscillatory time series in high-frequency finance and sensor applications.

Technology Category

• Machine Learning: Kernel Methods
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Data Mining & Knowledge Management: Anomaly/Outlier Detection

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• Extends signature kernel analysis to rough paths
• Develops novel PDE system for higher-order approximation
• Addresses computational intractability for oscillatory paths

Innovation

• Extends signature kernels to rough paths
• Introduces novel system of PDEs
• Higher order approximation with error bounds