Computational aspects of the Volterra Signature

TL;DR

This paper introduces efficient algorithms for computing the Volterra signature, a generalization of the path signature with matrix-valued kernels, addressing computational challenges and providing practical implementations.

Contribution

It develops novel algorithms with various complexity optimizations for the Volterra signature and releases an open-source package implementing these methods.

Findings

Quadratic complexity $O(J^2)$ for general approximation scheme

FFT-based acceleration reduces complexity to $O(J ext{log} J)$

Exact recursion achieves $O(JR^2)$ complexity for certain kernels

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".