# Global universality via discrete-time signatures

arXiv: https://arxiv.org/abs/2603.09773 · HTML: https://mlsanity.com/paper/2603.09773/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 38 references that are in the map.
- First posted on arXiv 10 Mar 2026.

## Builds on (most cited first, 15 of 15)

- [Kernels for sequentially ordered data](https://mlsanity.com/paper/1601.08169/) (2016)
- [Deep Signature Transforms](https://mlsanity.com/paper/1905.08494/) (2019)
- [Signature moments to characterize laws of stochastic processes](https://mlsanity.com/paper/1810.10971/) (2018)
- [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)
- [Embedding and learning with signatures](https://mlsanity.com/paper/1911.13211/) (2019)
- [Discrete-time signatures and randomness in reservoir computing](https://mlsanity.com/paper/2010.14615/) (2020)
- [Global universal approximation of functional input maps on weighted spaces](https://mlsanity.com/paper/2306.03303/) (2023)
- [Sparse arrays of signatures for online character recognition](https://mlsanity.com/paper/1308.0371/) (2013)
- [Persistence paths and signature features in topological data analysis](https://mlsanity.com/paper/1806.00381/) (2018)
- [Signature Methods in Stochastic Portfolio Theory](https://mlsanity.com/paper/2310.02322/) (2023)
- [Stochastic Control with Signatures](https://mlsanity.com/paper/2406.01585/) (2024)
- [Time-warping invariants of multidimensional time series](https://mlsanity.com/paper/1906.05823/) (2019)
- [Global universal approximation with Brownian signatures](https://mlsanity.com/paper/2512.16396/) (2025)
- [Orthogonal polynomials on path-space](https://mlsanity.com/paper/2602.18808/) (2026)

## Abstract (arXiv)

We establish global universal approximation theorems for non-anticipative and general path-dependent functionals on spaces of piecewise linear paths, stating that linear functionals of the corresponding signatures are dense with respect to Lp- and weighted norms. We verify that these approximation results are applicable to piecewise linear interpolations of a class of Gaussian processes, including Brownian motion and fractional Brownian motion with Hurst parameter H>1/4. Moreover, we derive quantitative convergence rates for signatures of piecewise linear approximations of Gaussian processes towards their continuous-time counterparts. Consequently, we obtain Lp-approximation results for path-dependent functionals of Gaussian processes, as well as for random ordinary differential equations and stochastic differential equations driven by Brownian motion.