# New directions in the applications of rough path theory

arXiv: https://arxiv.org/abs/2302.04586 · HTML: https://mlsanity.com/paper/2302.04586/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 31 references that are in the map.
- First posted on arXiv 9 Feb 2023.

## Builds on (most cited first, 16 of 16)

- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Neural SDEs as Infinite-Dimensional GANs](https://mlsanity.com/paper/2102.03657/) (2021)
- [Neural Rough Differential Equations for Long Time Series](https://mlsanity.com/paper/2009.08295/) (2020)
- [A Primer on the Signature Method in Machine Learning](https://mlsanity.com/paper/1603.03788/) (2016)
- [Kernels for sequentially ordered data](https://mlsanity.com/paper/1601.08169/) (2016)
- [Deep Signature Transforms](https://mlsanity.com/paper/1905.08494/) (2019)
- [The Signature Kernel is the solution of a Goursat PDE](https://mlsanity.com/paper/2006.14794/) (2020)
- [Signature moments to characterize laws of stochastic processes](https://mlsanity.com/paper/1810.10971/) (2018)
- [Neural Controlled Differential Equations for Online Prediction Tasks](https://mlsanity.com/paper/2106.11028/) (2021)
- [Distribution Regression for Sequential Data](https://mlsanity.com/paper/2006.05805/) (2020)
- [Higher Order Kernel Mean Embeddings to Capture Filtrations of Stochastic Processes](https://mlsanity.com/paper/2109.03582/) (2021)
- [Framing RNN as a kernel method: A neural ODE approach](https://mlsanity.com/paper/2106.01202/) (2021)
- [SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data](https://mlsanity.com/paper/2105.04211/) (2021)
- [Policy Analysis using Synthetic Controls in Continuous-Time](https://mlsanity.com/paper/2102.01577/) (2021)

## Built on by (most cited first, 12 of 12)

- [Theoretical Foundations of Deep Selective State-Space Models](https://mlsanity.com/paper/2402.19047/) (2024)
- [Non-adversarial training of Neural SDEs with signature kernel scores](https://mlsanity.com/paper/2305.16274/) (2023)
- [Structured Linear CDEs: Maximally Expressive and Parallel-in-Time Sequence Models](https://mlsanity.com/paper/2505.17761/) (2025)
- [Exact Gradients for Stochastic Spiking Neural Networks Driven by Rough Signals](https://mlsanity.com/paper/2405.13587/) (2024)
- [pySigLib -- Fast Signature-Based Computations on CPU and GPU](https://mlsanity.com/paper/2509.10613/) (2025)
- [Rademacher Complexity of Neural ODEs via Chen-Fliess Series](https://mlsanity.com/paper/2401.16655/) (2024)
- [Novelty Detection on Radio Astronomy Data using Signatures](https://mlsanity.com/paper/2402.14892/) (2024)
- [ParallelFlow: Parallelizing Linear Transformers via Flow Discretization](https://mlsanity.com/paper/2504.00492/) (2025)
- [Continuously Evolving Graph Neural Controlled Differential Equations for Traffic Forecasting](https://mlsanity.com/paper/2401.14695/) (2024)
- [Path Signatures Enable Model-Free Mapping of RNA Modifications](https://mlsanity.com/paper/2511.08855/) (2025)
- [Log-PDE Methods for Rough Signature Kernels](https://mlsanity.com/paper/2404.02926/) (2024)
- [Novelty detection on path space](https://mlsanity.com/paper/2512.03243/) (2025)

## Abstract (arXiv)

This article provides a concise overview of some of the recent advances in the application of rough path theory to machine learning. Controlled differential equations (CDEs) are discussed as the key mathematical model to describe the interaction of a stream with a physical control system. A collection of iterated integrals known as the signature naturally arises in the description of the response produced by such interactions. The signature comes equipped with a variety of powerful properties rendering it an ideal feature map for streamed data. We summarise recent advances in the symbiosis between deep learning and CDEs, studying the link with RNNs and culminating with the Neural CDE model. We concluded with a discussion on signature kernel methods.