Computational aspects of the Volterra Signature

2026-05-18

AI Summary

This work addresses the high computational complexity of Volterra signatures with matrix-valued kernels in time series modeling by proposing three efficient algorithms: a general O(J²) approximation, an FFT-based O(J log J) acceleration, and an exact recursive O(JR²) method tailored for state-space kernels. The authors demonstrate that multi-factor kernels do not increase asymptotic complexity. By integrating Chen-type convolution factorization, fast Fourier transforms, state-space representations, and JAX-based automatic differentiation, the proposed methods enable efficient computation of Volterra signatures. All algorithms are implemented in the open-source JAX library “tensordev,” significantly reducing computational costs while preserving the standard complexity dependence on path dimension and truncation order.

Abstract

The Volterra signature extends the classical path signature by incorporating general matrix-valued kernel into its iterated integral structure, yielding a flexible notion of memory for time series. Its components can be viewed as successive Picard iterates of linear controlled Volterra equations, making their exact computation of additional mathematical interest. However, the kernel introduces substantial algorithmic challenges. We provide a resolution by first decomposing the Chen-type convolution relation established in [arXiv:2603.04525] into analytic and arithmetic parts, and then introducing several efficient algorithms: a general approximative scheme with quadratic complexity $O(J^2)$ in the number of time steps $J$, an FFT-based acceleration with complexity $O(J\log J)$ for convolution kernels on uniform grids, and an exact recursion with complexity $O(JR^2)$ for kernels admitting a state-space representation of dimension $R$; retaining standard signature complexity in the path dimension and truncation level $N$. We further show that the number of factors in matrix-valued kernels of the form $K(t,s)=\sum_p k_p(t-s)A_p$ do not increase the asymptotic complexity in $J$ and $N$. Finally, we derive a finite-difference predictor--corrector scheme for the associated Volterra signature kernel. All algorithms are implemented in the publicly available JAX-based package "tensordev".

Problem

• Volterra signature
• matrix-valued kernel
• computational complexity
• iterated integrals
• time series memory

Innovation

• Volterra signature
• efficient algorithms
• state-space representation
• FFT acceleration
• Chen-type convolution