# Lecture notes on rough paths and applications to machine learning

arXiv: https://arxiv.org/abs/2404.06583 · HTML: https://mlsanity.com/paper/2404.06583/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 97 references that are in the map.
- First posted on arXiv 9 Apr 2024.

## Builds on (most cited first, 12 of 12)

- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [Neural SDEs as Infinite-Dimensional GANs](https://mlsanity.com/paper/2102.03657/) (2021)
- [Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU](https://mlsanity.com/paper/2001.00706/) (2020)
- [The iisignature library: efficient calculation of iterated-integral signatures and log signatures](https://mlsanity.com/paper/1802.08252/) (2018)
- [Metrizing Weak Convergence with Maximum Mean Discrepancies](https://mlsanity.com/paper/2006.09268/) (2020)
- [Higher Order Kernel Mean Embeddings to Capture Filtrations of Stochastic Processes](https://mlsanity.com/paper/2109.03582/) (2021)
- [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)
- [Signature Kernel Conditional Independence Tests in Causal Discovery for Stochastic Processes](https://mlsanity.com/paper/2402.18477/) (2024)
- [SK-Tree: a systematic malware detection algorithm on streaming trees via the signature kernel](https://mlsanity.com/paper/2102.07904/) (2021)
- [A Neural RDE approach for continuous-time non-Markovian stochastic control problems](https://mlsanity.com/paper/2306.14258/) (2023)

## Built on by (most cited first, 14 of 14)

- [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)
- [Sparse Signature Coefficient Recovery via Kernels](https://mlsanity.com/paper/2412.08579/) (2024)
- [Universal randomised signatures for generative time series modelling](https://mlsanity.com/paper/2406.10214/) (2024)
- [SigDiffusions: Score-Based Diffusion Models for Time Series via Log-Signature Embeddings](https://mlsanity.com/paper/2406.10354/) (2024)
- [ParallelFlow: Parallelizing Linear Transformers via Flow Discretization](https://mlsanity.com/paper/2504.00492/) (2025)
- [Infinite-dimensional Mahalanobis Distance with Applications to Kernelized Novelty Detection](https://mlsanity.com/paper/2407.11873/) (2024)
- [Local regression on path spaces with signature metrics](https://mlsanity.com/paper/2510.16728/) (2025)
- [Autonomous-Flow-Based Generation](https://mlsanity.com/paper/2511.09902/) (2025)
- [Conservative Continuous-Time Treatment Optimization](https://mlsanity.com/paper/2603.16789/) (2026)
- [Detecting malignant dynamics on very few blood sample using signature coefficients](https://mlsanity.com/paper/2506.09097/) (2025)
- [Random Controlled Differential Equations](https://mlsanity.com/paper/2512.23670/) (2025)
- [Advances in Neural Controlled Differential Equations](https://mlsanity.com/paper/2607.05280/) (2026)
- [Time series generation with spectrally aligned latent flow matching](https://mlsanity.com/paper/2609.21989/) (2026)

## Abstract (arXiv)

These notes expound the recent use of the signature transform and rough path theory in data science and machine learning. We develop the core theory of the signature from first principles and then survey some recent popular applications of this approach, including signature-based kernel methods and neural rough differential equations. The notes are based on a course given by the two authors at Imperial College London.