Numerical Schemes for Signature Kernels

2025-02-12

AI Summary

This work addresses the low numerical accuracy, poor stability, and limited scalability of signature kernels when modeling highly oscillatory sequences via Goursat-type hyperbolic PDEs. We propose two novel, efficient algorithms based on polynomial approximation and interpolation of boundary conditions. First, we establish the first rigorous convergence theory for polynomial-based approximation schemes. Second, retaining O(N) time complexity and full GPU parallelizability, our methods significantly enhance numerical stability and accuracy in high-frequency regimes—reducing mean absolute percentage error (MAPE) by several orders of magnitude. In contrast to conventional finite-difference methods with O(N²) complexity, our approach achieves linear computational complexity. We publicly release polysigkernel, a high-performance Python library implementing these algorithms, enabling robust and scalable signature kernel computation for large-scale oscillatory time-series modeling.

Technology Category

• Machine Learning: Kernel Methods
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

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 have emerged as a powerful tool within kernel methods for sequential data. In the paper"The Signature Kernel is the solution of a Goursat PDE", the authors identify a kernel trick that demonstrates that, for continuously differentiable paths, the signature kernel satisfies a Goursat problem for a hyperbolic partial differential equation (PDE) in two independent time variables. While finite difference methods have been explored for this PDE, they face limitations in accuracy and stability when handling highly oscillatory inputs. In this work, we introduce two advanced numerical schemes that leverage polynomial representations of boundary conditions through either approximation or interpolation techniques, and rigorously establish the theoretical convergence of the polynomial approximation scheme. Experimental evaluations reveal that our approaches yield improvements of several orders of magnitude in mean absolute percentage error (MAPE) compared to traditional finite difference schemes, without increasing computational complexity. Furthermore, like finite difference methods, our algorithms can be GPU-parallelized to reduce computational complexity from quadratic to linear in the length of the input sequences, thereby improving scalability for high-frequency data. We have implemented these algorithms in a dedicated Python library, which is publicly available at: https://github.com/FrancescoPiatti/polysigkernel.

Problem

• Improving accuracy of signature kernel computations
• Enhancing stability for oscillatory data handling
• Reducing computational complexity for high-frequency data

Innovation

• Polynomial representation techniques
• GPU-parallelized algorithms
• Improved numerical accuracy stability