# Universal approximation with signatures of non-geometric rough paths

arXiv: https://arxiv.org/abs/2602.05898 · HTML: https://mlsanity.com/paper/2602.05898/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 48 references that are in the map.
- First posted on arXiv 5 Feb 2026.

## Builds on (most cited first, 5 of 5)

- [Signature Methods in Machine Learning](https://mlsanity.com/paper/2206.14674/) (2022)
- [Signature Methods in Stochastic Portfolio Theory](https://mlsanity.com/paper/2310.02322/) (2023)
- [Applications of Signature Methods to Market Anomaly Detection](https://mlsanity.com/paper/2201.02441/) (2022)
- [On Consistency of Signature Using Lasso](https://mlsanity.com/paper/2305.10413/) (2023)
- [Signature Trading: A Path-Dependent Extension of the Mean-Variance Framework with Exogenous Signals](https://mlsanity.com/paper/2308.15135/) (2023)

## Built on by (most cited first, 4 of 4)

- [Deep Hilbert--Galerkin Methods for Infinite-Dimensional PDEs and Optimal Control](https://mlsanity.com/paper/2603.19463/) (2026)
- [Universal Approximation of Nonlinear Operators and Their Derivatives](https://mlsanity.com/paper/2605.15285/) (2026)
- [Branched Signature Kernel Solvers for ODEs with rough Single-Trajectory signals](https://mlsanity.com/paper/2605.25826/) (2026)
- [Faithful Embeddings of Irregular and Asynchronous Data for Online Log-NCDEs](https://mlsanity.com/paper/2605.30213/) (2026)

## Abstract (arXiv)

We establish a universal approximation theorem for signatures of rough paths that are not necessarily weakly geometric. By extending the path with time and its rough path bracket terms, we prove that linear functionals of the signature of the resulting rough paths approximate continuous functionals on rough path spaces uniformly on compact sets. Moreover, we construct the signature of a path extended by its pathwise quadratic variation terms based on general pathwise stochastic integration `a la F"ollmer, in particular, allowing for pathwise Ito, Stratonovich, and backward Ito integration. In a probabilistic setting, we obtain a universal approximation result for linear functionals of the signature of continuous semimartingales extended by the quadratic variation terms, defined via stochastic Ito integration. Numerical examples illustrate the use of signatures when the path is extended by time and quadratic variation in the context of model calibration and option pricing in mathematical finance.