# The Volterra signature

arXiv: https://arxiv.org/abs/2603.04525 · HTML: https://mlsanity.com/paper/2603.04525/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 84 references that are in the map.
- First posted on arXiv 4 Mar 2026.

## Builds on (most cited first, 21 of 21)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Linformer: Self-Attention with Linear Complexity](https://mlsanity.com/paper/2006.04768/) (2020)
- [Transformers are SSMs: Generalized Models and Efficient Algorithms Through Structured State Space Duality](https://mlsanity.com/paper/2405.21060/) (2024)
- [The Mythos of Model Interpretability](https://mlsanity.com/paper/1606.03490/) (2016)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [The UEA multivariate time series classification archive, 2018](https://mlsanity.com/paper/1811.00075/) (2018)
- [A Primer on the Signature Method in Machine Learning](https://mlsanity.com/paper/1603.03788/) (2016)
- [Kernels for sequentially ordered data](https://mlsanity.com/paper/1601.08169/) (2016)
- [The Signature Kernel is the solution of a Goursat PDE](https://mlsanity.com/paper/2006.14794/) (2020)
- [The iisignature library: efficient calculation of iterated-integral signatures and log signatures](https://mlsanity.com/paper/1802.08252/) (2018)
- [Signature Methods in Machine Learning](https://mlsanity.com/paper/2206.14674/) (2022)
- [Conditional Sig-Wasserstein GANs for Time Series Generation](https://mlsanity.com/paper/2006.05421/) (2020)
- [Neural Integral Equations](https://mlsanity.com/paper/2209.15190/) (2022)
- [Feature Engineering with Regularity Structures](https://mlsanity.com/paper/2108.05879/) (2021)
- [Stochastic Control with Signatures](https://mlsanity.com/paper/2406.01585/) (2024)
- [Generalized iterated-sums signatures](https://mlsanity.com/paper/2012.04597/) (2020)
- [Hedging with memory: shallow and deep learning with signatures](https://mlsanity.com/paper/2508.02759/) (2025)
- [Neural stochastic Volterra equations: learning path-dependent dynamics](https://mlsanity.com/paper/2407.19557/) (2024)

## Built on by (most cited first, 5 of 5)

- [Deep Hilbert--Galerkin Methods for Infinite-Dimensional PDEs and Optimal Control](https://mlsanity.com/paper/2603.19463/) (2026)
- [Extending SSMs with the Exponentially Weighted Signature](https://mlsanity.com/paper/2603.19198/) (2026)
- [Adaptive Learning via Off-Model Training and Importance Sampling for Fully Non-Markovian Optimal Stochastic Control. Complete version](https://mlsanity.com/paper/2604.13147/) (2026)
- [Universal Approximation of Nonlinear Operators and Their Derivatives](https://mlsanity.com/paper/2605.15285/) (2026)
- [Computational aspects of the Volterra Signature](https://mlsanity.com/paper/2605.18406/) (2026)

## Abstract (arXiv)

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 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 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, 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.