The Volterra signature

2026-03-04

AI Summary

This work addresses the challenges of opaque memory mechanisms and difficulties in long-horizon training inherent in non-Markovian time series modeling by proposing Volterra signatures as an explicit representation of historical dependencies. These signatures are derived by expanding input paths with temporal kernels within a tensor algebra framework, and theoretical learning guarantees are established via the Volterra–Chen identity. Under augmented conditions, we prove the identifiability of these signatures and their universal approximation capability over infinite-dimensional path spaces—requiring only linear functionals in certain cases. Moreover, we show that their inner product admits a closed-form expression through a two-parameter integral equation, enabling compatibility with kernel methods and numerical PDE solvers. Experiments demonstrate that the proposed approach significantly outperforms classical path signature baselines on both real-world and synthetic dynamical tasks, achieving superior expressiveness while maintaining computational feasibility.

Abstract

Modern approaches for learning from non-Markovian time series, such as recurrent neural networks, neural controlled differential equations or transformers, typically rely on implicit memory mechanisms that can be difficult to interpret or to train over long horizons. We propose the Volterra signature $\mathrm{VSig}(x;K)$ as a principled, explicit feature representation for history-dependent systems. By developing the input path $x$ weighted by a temporal kernel $K$ into the tensor algebra, we leverage the associated Volterra--Chen identity to derive rigorous learning-theoretic guarantees. Specifically, we prove an injectivity statement (identifiability under augmentation) that leads to a universal approximation theorem on the infinite dimensional path space, which in certain cases is achieved by linear functionals of $\mathrm{VSig}(x;K)$. Moreover, we demonstrate applicability of the kernel trick by showing that the inner product associated with Volterra signatures admits a closed characterization via a two-parameter integral equation, enabling numerical methods from PDEs for computation. For a large class of exponential-type kernels, $\mathrm{VSig}(x;K)$ solves a linear state-space ODE in the tensor algebra. Combined with inherent invariance to time reparameterization, these results position the Volterra signature as a robust, computationally tractable feature map for data science. We demonstrate its efficacy in dynamic learning tasks on real and synthetic data, where it consistently improves classical path signature baselines.

Problem

• non-Markovian time series
• implicit memory mechanisms
• interpretability
• long-horizon training
• history-dependent systems

Innovation

• Volterra signature
• tensor algebra
• universal approximation
• kernel trick
• non-Markovian time series