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Lecture notes on rough paths and applications to machine learning

Thomas Cass, Cristopher Salvi

TL;DR

The notes develop the signature transform as a robust, path-space representation built from iterated integrals and encoded via controlled differential equations. They establish a comprehensive algebraic and analytic framework, including Chen's relation and the shuffle identity, and define unparameterised path spaces with a topology suitable for learning. The composition with kernel methods yields universal, characteristic signature kernels and RKHSs, enabling regression, distribution regression, and hypothesis testing on path-valued data, with practical computation via PDE-based schemes and truncations. Finally, the notes bridge to rough path theory and deep learning, outlining neural rough differential equations and their training, thereby linking classical stochastic analysis with contemporary data-driven sequence modeling. The material provides both theoretical foundations and practical tools for applying signatures in time-series analysis, kernel learning, and neural differential equations. Practically, this framework supports efficient, invariant representations of streamed data and principled, scalable learning pipelines for sequential tasks.

Abstract

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.

Lecture notes on rough paths and applications to machine learning

TL;DR

The notes develop the signature transform as a robust, path-space representation built from iterated integrals and encoded via controlled differential equations. They establish a comprehensive algebraic and analytic framework, including Chen's relation and the shuffle identity, and define unparameterised path spaces with a topology suitable for learning. The composition with kernel methods yields universal, characteristic signature kernels and RKHSs, enabling regression, distribution regression, and hypothesis testing on path-valued data, with practical computation via PDE-based schemes and truncations. Finally, the notes bridge to rough path theory and deep learning, outlining neural rough differential equations and their training, thereby linking classical stochastic analysis with contemporary data-driven sequence modeling. The material provides both theoretical foundations and practical tools for applying signatures in time-series analysis, kernel learning, and neural differential equations. Practically, this framework supports efficient, invariant representations of streamed data and principled, scalable learning pipelines for sequential tasks.

Abstract

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.
Paper Structure (59 sections, 58 theorems, 342 equations, 3 figures)

This paper contains 59 sections, 58 theorems, 342 equations, 3 figures.

Table of Contents

  1. The signature transform
  2. Controlled differential equations
  3. Geometric view
  4. Coordinate iterated integrals
  5. Algebras of tensors
  6. Inner products
  7. Functions on the range of the signature transform
  8. Analytical properties
  9. Sampling invariance
  10. Fast decay in the magnitude of coefficients
  11. Uniqueness
  12. Algebraic properties
  13. Chen's relation
  14. The signature as a controlled differential equation
  15. The shuffle identity
  16. ...and 44 more sections

Key Result

Lemma 1.1.16

The vector spaces $T\left( \left( V^{\ast }\right) \right)$ and $T\left( V\right) ^{\ast \ }$can be identified through the explicit isomorphism $\Phi :T\left( \left( V^{\ast }\right) \right) \rightarrow T\left( V\right) ^{\ast \ }$ where, for any $v=\sum_{k\in A}v_{k}\in T\left( V\right)$ with $A\subset \mathbb{N}_{0}$ a finite subset, $\Phi _{f}\left( v\right)$ is defined to equal $\sum_{k\in A}f

Figures (3)

  • Figure 1: On the left are the individual channels $(\gamma^x, \gamma^y)$ of a $2$D paths $\gamma$. In the middle are the channels reparametrized under $\phi: t \mapsto t^2$. On the right are the path $\gamma$ and its reparametrized version $\gamma \circ \phi$. The two curves overlap, meaning that the reparametrization $\phi$ represents irrelevant information if one is interested in understanding the shape of $\gamma$. Figure taken from salvi2021signature.
  • Figure 2: Distinct planar axis paths having the same signatures when truncated at levels one and two respectively (left to right).
  • Figure 3: Simulation of the trajectories traced by $20$ particles of an ideal gas in a $3$-d box under different thermodynamic conditions. Higher temperatures equate to a higher internal energy in the system which increases the number of collisions resulting in different large-scale dynamics of the gas. Figure taken from lemercier2021distribution.

Theorems & Definitions (172)

  • Definition 1.1.1: Vector field
  • Definition 1.1.2
  • Definition 1.1.3
  • Definition 1.1.4: $p$-variation
  • Definition 1.1.6: Signature transform
  • Remark 1.1.7
  • Definition 1.1.8
  • Definition 1.1.9: Extended tensor algebra
  • Remark 1.1.10: Tensor algebra
  • Definition 1.1.11
  • ...and 162 more