arXiv subjects

Fabian N. Harang

Publications and source records attributed to Fabian N. Harang.

3 recordsLinked to original sources

arXiv2026

Dynamic Universal Approximation via Signature Controlled Differential Equations

We study signature controlled differential equations (Sig-CDEs), that is, path-dependent controlled differential equations (CDEs) whose vector fields factor through the signature map. Working on spaces of stopped Hölder paths, we develop an existence, uniqueness, and stability theory for general path-dependent CDEs, and translate these pathwise well-posedness criteria into conditions on the corresponding signature functionals. We then prove dynamic universality: simply parametrized Sig-CDEs approximate the solution path of any well-posed path-dependent CDE arbitrarily well, uniformly over bounded sets of controls and initial histories, with global variants obtained using weighted spaces. Within this framework, entire maps of group-like elements provide a specific class of Sig-CDEs. Using a new class of limiting tensor spaces, we recast Sig-CDEs as infinite-dimensional classical CDEs and prove their well-posedness via a gauge-type scaling argument, thereby establishing a principled way to lift generic path-dependent dynamics. Lastly, we study truncated Sig-CDEs as finite-dimensional differential equations on step$-N$ Lie groups under intrinsic conditions, that is, with well-posedness formulated in terms of the underlying group metric.

math.CA↗

arXiv2026

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 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.

stat.ML↗

arXiv2026

Computational aspects of the Volterra Signature

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".

math.NA↗