The Volterra signature

Paul P. Hager; Fabian N. Harang; Luca Pelizzari; Samy Tindel

stat.MLMar 4, 2026

The Volterra signature

Authors: Paul P. Hager, Fabian N. Harang, Luca Pelizzari, Samy Tindel

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.

Figures & tables

Figure 4.1 . Volterra signature vs. classical signature expansions, compared with the fractional SDE solution ( 4.1 ) for one testing sample. The models were trained with

M=900

training samples and

N=500

time-steps on

[0,1]

Parameters:

Y_{0}=1,b_{0}=0,b_{1}=-1,\sigma_{0}=1,\sigma_{1}=0.5

, signature truncation

L=6

Method	$\mathrm{MSE}\!\mid_{[0,1]}$	$R^{2}\!\mid_{[0,1]}$	$\mathrm{MSE}\!\mid_{[0,2]}$	$R^{2}\!\mid_{[0,2]}$
Sig	$2.29\times 10^{-4}$	1.000	$2.9387$	-0.923
$\texttt{VSig}_{k}$	$6.30\times 10^{-5}$	1.000	$0.0012$	0.999
$\texttt{VSig}_{k_{\lambda}}$	$1.38\times 10^{-4}$	1.000	$0.0124$	0.992

Table 1. Prediction performance of the three feature maps Sig ,

\texttt{VSig}_{k}

, and

\texttt{VSig}_{k_{\lambda}}

, evaluated on the intervals

[0,1]

and

[0,2]

. We report the mean squared error (MSE) and the coefficient of determination

R^{2}

Figure 4.2 . Next-day forecast of realized S&P 500 volatility using our method VSig , compared with the benchmark HAR , on the test set. The lower subplot shows the absolute forecast errors

|\widehat{y}-y|

for both methods, where

y

denotes the realized volatility.

Figure 4.3 . Left: Coefficient of determination

R^{2}

as a (linearly interpolated) function of the past-window size

p

(days), reported on the training and test sets for the methods VSig and Sig , and the (constant) benchmark HAR . Right: Scatter plots of realized volatility

y

versus predictions

\widehat{y}

for our method VSig and the HAR benchmark.

	20	60	110	150	200	240
q
1	0.59/ 0.60 /0.60	0.58/ 0.62 /0.60	0.57/ 0.63 /0.60	0.56/ 0.64 /0.60	0.52/ 0.65 /0.60	0.47/ 0.65 /0.60
3	0.34/0.33/0.38	0.38/ 0.38 /0.38	0.38/ 0.39 /0.38	0.39/ 0.40 /0.38	0.36/ 0.40 /0.38	0.35/ 0.42 /0.38
5	0.22 /0.21/0.25	0.27 /0.24/0.25	0.29 /0.25/0.25	0.31 /0.25/0.25	0.28 /0.26/0.25	0.27/ 0.28 /0.25

Table 2.

R^{2}

overview on the test set: ( Sig / VSig / HAR ) for forecasting horizons

q\in\{1,3,5\}

and several past-window sizes

p

Datasets/Kernels	Linear	RBF	GAK	Sig	VSig	Sig-RBF	VSig-RBF
ArticularyWordRecognition	98.0	98.0	98.0	92.3	98.7	98.3	99.0
BasicMotions	87.5	97.5	97.5	97.5	97.5	100.0	97.5
Cricket	91.7	91.7	97.2	86.1	90.3	97.2	97.2
Libras	73.9	77.2	79.0	81.7	87.8	81.7	86.7
NATOPS	90.0	92.2	90.6	88.3	93.9	93.3	92.2
RacketSports	76.9	78.3	84.2	80.2	73.0	84.9	90.8

Table 3. Classification accuracies (%) on UCR/UEA datasets. The best test accuracy in each row is highlighted in bold.

Path Signatures

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Mar 19, 2026

Extending SSMs with the Exponentially Weighted Signature

We introduce the exponentially weighted signature (EWS), a continuous-time model that computes iterated integrals of a path, where each increment is weighted by the matrix exponential of a learnable generator over elapsed clock time. We prove that it solves a linear controlled differential equation, keeps the group-like structure and the universality of the signature, and satisfies a modified Chen identity, enabling a parallel scan. At depth one the EWS is a state-space model (SSM), and we map linear time-invariant SSMs, Mamba channels and Mamba-

2

heads to it in closed form. The EWS extends SSMs through an arbitrary matrix generator, a clock that generalises the step size to causal functionals of the input, and higher truncation depths that are non-linear in the path within a single layer. Empirically, the EWS achieves the highest average accuracy and rank on six long time-series classification datasets, where depth generally helps. Learned clocks prove necessary for state tracking on formal language tasks, and at depth one, the EWS matches or exceeds competing SSMs on regression and forecasting with far fewer parameters.

Alexandre Bloch, Benjamin Walker, Joël Mouterde +3

Mathematical Institute, University of Oxford · SKF · Department of Mathematics, Imperial College London

Dynamical Systems Time Series Classification Time Series Forecasting State Space Models

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Jul 31, 2025

Signature-Kernel One-Class Learning with Manifold Regularisation for Multivariate Time Series Anomaly Detection

One-class learning is an effective methodology for time series anomaly detection, yet kernel one-class approaches often compare sequences in a pointwise fashion and ignore the geometry of normal data. In this work, we present a manifold-regularised, large-margin

\ell_p

-support vector data description (

\ell_p

-SVDD) method that analyses sequences as paths via the signature kernel. By incorporating a a graph-Laplacian penalty into the

\ell_p

-SVDD objective function and utilising the representer theorem for manifold regularisation, we illustrate that the dual task retains the functional form of the original one with a modified kernel matrix. As such, existing Frank--Wolfe solvers can be applied unchanged. We show that this new modified kernel is a valid kernel matrix and under a generic norm bound on the discriminant, the proposed model yields a smaller Rademacher-complexity bound than the baseline. Furthermore, the signature kernel which is computed through a Goursat partial differential equation and can handle irregularly sampled data and is invariant to reparametrisation has not previously been used for one-class time series anomaly detection to the best of our knowledge. Our experiments on five widely used benchmarks, evaluated without point adjustment, demonstrate that the proposed approach compares very favourably with the state of the art approaches.

Shervin Rahimzadeh Arashloo

Department of Computer Engineering, Faculty of Engineering, Bilkent University, Ankara, Turkey.

Time-Series Anomaly Detection One-Class Classification Kernel Methods

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Aug 22, 2026

Symbolic Neural ODEs: Learning interpretable models from time-series data

We present a machine learning framework for identifying sparse, interpretable models of dynamical systems directly from time-series data. Our approach parameterizes the underlying vector field using a neural architecture and trains it by minimizing a multi-step prediction loss over a finite horizon. To ensure numerical tractability, we optimize a mean absolute error objective averaged across prediction steps, and progressively increase the horizon during training. A key feature of this formulation is that it enforces consistency under repeated composition of the learned dynamics. As a result, the identified models exhibit significantly improved stability compared with approaches based on one-step regression of the vector field. When combined with sparsity-promoting regularization, this leads to parsimonious models that generalize beyond the training data. We demonstrate accurate recovery of systems exhibiting a wide range of behaviors, including stable and unstable fixed points, periodic orbits, and chaotic attractors. For chaotic systems, while long-term trajectory prediction is inherently limited by sensitivity to initial conditions, we show that multi-step training yields models with accurate short-term dynamics and strong agreement in long-time statistical properties, including mean, variance, and Lyapunov exponents. Moreover, we establish theoretical bounds linking trajectory error to statistical accuracy, providing a step toward a principled explanation for this behavior.

Nibodh Boddupalli, Jeff Moehlis

Dept. of Mechanical Engineering, University of California, Santa Barbara, CA 93106

Nonlinear System Identification System Identification Neural ODEs Sparse Identification of Nonlinear Dynamics Ordinary Differential Equations Chaotic Time-Series Prediction

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Abstract

Path Signatures

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