Scalable Signature Kernel Computations for Long Time Series via Local Neumann Series Expansions

2025-02-27

AI Summary

Addressing the longstanding trade-off between accuracy and computational efficiency in computing signature kernels for long, high-dimensional time series, this paper introduces a dynamic truncation local Neumann series method grounded in Goursat-type partial differential equations. The approach employs tilewise domain decomposition, a directed-graph topological ordering to guide boundary propagation, and adaptive precision truncation, enabling recursive construction of rapidly convergent power series approximations over subregions. This is the first method to enable high-accuracy signature kernel computation on sequences exceeding 500,000 points using a single GPU—reducing memory consumption by over an order of magnitude. Notably, it achieves significantly improved accuracy on highly rough (low-regularity) paths. These advances substantially enhance the practical applicability of rough path theory in real-world domains such as finance and signal processing.

Technology Category

• Machine Learning: Kernel Methods
• Data Mining & Knowledge Management: Data Compression
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

The signature kernel is a recent state-of-the-art tool for analyzing high-dimensional sequential data, valued for its theoretical guarantees and strong empirical performance. In this paper, we present a novel method for efficiently computing the signature kernel of long, high-dimensional time series via dynamically truncated recursive local power series expansions. Building on the characterization of the signature kernel as the solution of a Goursat PDE, our approach employs tilewise Neumann-series expansions to derive rapidly converging power series approximations of the signature kernel that are locally defined on subdomains and propagated iteratively across the entire domain of the Goursat solution by exploiting the geometry of the time series. Algorithmically, this involves solving a system of interdependent local Goursat PDEs by recursively propagating boundary conditions along a directed graph via topological ordering, with dynamic truncation adaptively terminating each local power series expansion when coefficients fall below machine precision, striking an effective balance between computational cost and accuracy. This method achieves substantial performance improvements over state-of-the-art approaches for computing the signature kernel, providing (a) adjustable and superior accuracy, even for time series with very high roughness; (b) drastically reduced memory requirements; and (c) scalability to efficiently handle very long time series (e.g., with up to half a million points or more) on a single GPU. These advantages make our method particularly well-suited for rough-path-assisted machine learning, financial modeling, and signal processing applications that involve very long and highly volatile data.

Problem

• Efficient computation of signature kernel
• Handling long high-dimensional time series
• Local Neumann series expansions for scalability

Innovation

• Local Neumann series expansions
• Dynamic truncation technique
• Scalable GPU implementation