Rough paths: machine learning for sequential data

Explaining the science

The Science Behind Rough Path Theory 

RPT was developed in the 1990s by Terry Lyons with the initial goal of making precise the mechanism of how irregular signals interact with non-linear systems. It provides a deterministic toolbox to recover many classical results in stochastic analysis without using specific probabilistic arguments. In particular, it extends Itô's theory of SDEs far beyond the semimartingale setting. At the heart of it is the challenge of developing a robust solution theory for controlled differential equations driven by very irregular signals. 
 
One of its core tools, the signature, is a homomorphism from the monoid of paths with concatenation into the grouplike elements embedded into the tensor algebra; it provides a graded and faithful description of a path (up to appropriate reparameterizations) by locally removing the need to look at its fine structure and summarising it over short intervals. Linear functionals on the signature form a unital algebra (with the shuffle product) that separates points, therefore they form a basis for continuous functions on compact sets of paths. In practice, this translates into the following - almost magical - fact: the problem of learning a complex, highly non-linear function on a dataset of irregular time series can be replaced by a simple, two-steps procedure: 1) extract features from the stream by computing its truncated signature and 2) perform linear regression on the signature features. 
 
RPT had many recent, exciting research spin-offs. On the theoretical side, Martin Hairer generalized RPT to construct a robust solution theory for certain classes of ill-posed stochastic PDEs - known as the theory of regularity structures - for which he was awarded a Fields medal in 2014. On the applied side, RPT has enabled the development of various learning tools - such as the signature kernel and the neural controlled/rough differential equation models - which are emerging are leading machine learning tools for time series data.