The signature of iterated integrals: algebra, analysis and machine learning

  • Lecturer: Joscha Diehl
  • Date: Thursday, 09.00-10.30
  • Room: A3 02
  • Language: English
  • Target audience: MSc studens, PhD students, Postdocs
  • Prerequisites: Basic knowledge of real analysis
  • Lecture topics in detail: see Personal Homepage of Joscha Diehl

Abstract:

The signature of iterated integrals was introduced by Chen in the early part of the 20th century. It maps smooth enough curves $$X: [0,T] \rightarrow \mathbb{R}^d$$ into an infinite collection of numbers
  \[
    \int_0^T \int_0^{r_n} \int_0^{r_2} dX^{i_1}_{r_1} .. dX^{i_n}_{r_n}.
  \]

We will cover

  • How analytic properties of integration translate to algebraic properties of the signature. This opens the door to studying the free associative algebra, the free Lie algebra and related Hopf algebraic concepts.
  • How X is (almost) completely determined by its signature.
  • How the signature has found application in stochastic analysis, via the theory of rough paths.
  • How the signature is used in machine learning as a means for feature extraction.

Regular Lectures (Summer 2018)

06.04.2018, 12:21