Applications of Rough Paths: Computational Signatures and Data Science

Rough path theory emerged as a branch of stochastic analysis to give an improved approach to dealing with the interactions of complex random systems. In that context, it continues to resolve important questions, but its broader theoretical footprint has been substantial. Most notable is its contribution to Hairer’s Fields-Medal-winning work on regularity structures. At the core of rough path theory is the so-called signature transform which, while being simple to define, has rich mathematical properties bringing in aspects of analysis, geometry, and algebra. Hambly and Lyons (Annals of Math, 2010) built upon earlier work of Chen, showing how the signature represents the path uniquely up to generalized reparameterizations. This turns out to have practical implications allowing one to summarise the space of functions on unparameterized paths and data streams in a very economical way.

Over the past five years, a significant strand of applied work has been undertaken to exploit the mathematical richness of this object in diverse data science challenges from healthcare, to computer vision to gesture recognition. The log signature is becoming a powerful way to summarise the fine structure of a data stream in a neural net. The emergence of neural differential equations as an important tool in data science further deepens the connections with rough paths.