Computational aspects of the Volterra Signature

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 \emph{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 \emph{injectivity} statement (identifiability under augmentation) that leads to a \emph{universal approximation} theorem on the infinite dimensional path space, which in certain cases is achieved by \emph{linear functionals} of $\mathrm{VSig}(x;K)$. Moreover, we demonstrate applicability of the \emph{kernel trick} by showing that the inner product associated with Volte …

Signature stochastic differential equations (SDEs) constitute a large class of stochastic processes, here driven by Brownian motions, whose characteristics are linear maps of their own signature, i.e. of iterated integrals of the process with itself, and therefore allow for a generic path dependence. We show that their prolongation with the corresponding signature is an affine and polynomial process taking values in the set of group-like elements of the extended tensor algebra. By relying on duality theory for affine or polynomial processes, we obtain explicit formulas in terms of converging power series for the Fourier-Laplace transform and the expected value of entire functions of the signature process’ marginals. The coefficients of these power series are solutions of Riccati and linear ordinary differential equations (ODEs) with values in the extended tensor algebra, respectively, whose vector fields can be expressed in terms of the characteristics of the corresponding SDEs. We thu …

This article is devoted to the extension of the theory of rough paths in the context of Volterra equations with possibly singular kernels. We begin to describe a class of two parameter functions defined on the simplex called Volterra paths. These paths are used to construct a so-called Volterra-signature, analogously to the signature used in Lyon's theory of rough paths. We provide a detailed algebraic and analytic description of this object. Interestingly, the Volterra signature does not have a multiplicative property similar to the classical signature, and we introduce an integral product behaving like a convolution extending the classical tensor product. We show that this convolution product is well defined for a large class of Volterra paths, and we provide an analogue of the extension theorem from the theory of rough paths (which guarantees in particular the existence of a Volterra signature). Moreover the concept of convolution product is essential in the construction of Volterra …

This paper deals with non-Lipschitz nonlinear systems. Such systems can be approximated by a linear map of so-called signatures, which play a crucial role in the theory of rough paths and can be interpreted as collections of iterated integrals involving the control process. As a consequence, we identify a universal bilinear system, solved by the signature, that can approximate the state or output of the original nonlinear dynamics arbitrarily well. In contrast to other (bi)linearization techniques, the signature approach remains feasible in large-scale settings, as the dimension of the associated bilinear system grows only with the number of inputs. However, the signature model is typically of high order, requiring an optimization process based on model order reduction (MOR). We derive an MOR method for unstable bilinear systems with non-zero initial states and apply it to the signature, yielding a potentially low-dimensional bilinear model. An advantage of our method is that the origi …

Signature kernels, inner products of path signatures, underpin several machine learning algorithms for multivariate time series analysis. For bounded variation paths, signature kernels were recently shown to solve a Goursat PDE. However, existing PDE solvers only use increments as input data, leading to first order approximation errors. These approaches become computationally intractable for highly oscillatory input paths, as they have to be resolved at a fine enough scale to accurately recover their signature kernel, resulting in significant time and memory complexities. In this paper, we extend the analysis to rough paths, and show, leveraging the framework of smooth rough paths, that the resulting rough signature kernels can be approximated by a novel system of PDEs whose coefficients involve higher order iterated integrals of the input rough paths. We show that this system of PDEs admits a unique solution and establish quantitative error bounds yielding a higher order approximation …