The Signature Transform in Numerics and Machine Learning

Abstract

In this work we study the <em>signature transform</em> from the viewpoint of applied and numerical mathematics. <br /> The theoretical background is established in the first part, where the signature is defined as a map going from continuous paths of bounded variations to ordered tensor algebras. Approximation theorems and computational considerations are clarified, together with explicit and well commented examples. Only selected essential properties are pointed out, useful for non-linear approximation of functionals, dimension reduction and extension to the probabilistic setting. <br /> In the second part we use all the previously introduced theory to design numerical experiments of interest in data science and machine learning, targeting problems like time series classification, clustering, correlation detection and generation of artificial samples. A small section on agents classification for reinforcement learning is also included. <br /> Finally, the reader is given a list of possible connections to other areas of mathematics like PDE, kernel theory, jump processes and even algebraic geometry. We did our best to keep the exposition clear and compact.