New directions in the applications of rough path theory

TL;DR

This paper reviews recent advances in applying rough path theory, especially controlled differential equations and signatures, to machine learning, highlighting their integration with deep learning models like Neural CDEs.

Contribution

It provides a concise overview of how rough path theory and signature methods are advancing machine learning, including connections with neural networks and kernel methods.

Findings

Signatures serve as powerful feature maps for streamed data.

Deep learning models like Neural CDEs effectively incorporate rough path concepts.

Signature kernel methods enhance machine learning applications.

Abstract

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.