Terry Lyons, Xin Zhang, Joscha Diehl and Josef Teichmann
Rough path theory in machine learning
In the era of ‘big data’, numerous data streams are being generated by individual activity, business processes and sensors. Data streams record complex sequences of events and are ubiquitous in everyday life; examples range from electronic financial trading records to human-computer interactions. Statistics and now machine learning have achieved considerable success in working with multimodal data streams. Rough Path Theory (RPT) provide a mathematical approach to the description of complex data streams; an approach that can be efficient, concise, robust to different sampling, assimilate new asynchronous features and is able to accommodate missing data. There are many contexts where incorporating these tools into the data analysis leads to improved efficiency (sometimes by orders of magnitude), quicker training and reduced power consumption. RPT originated as a branch of stochastic analysis. It provides a rigorous framework for the analysis of controlled differential equations (CDEs) driven by highly oscillatory streams. Linear functionals on CDEs are a universally approximating family of functions on streams, that allows simple parameterization and is closed under concatenation. It is well adapted to modern frameworks of data science (optimization, back-propagation) in addition to the intrinsic ability to deal with evolving multi-modal data. There are many exciting research questions and applications (eg algebraic methods providing a new structure theorem for information contained in streamed data, signature-based deep learning methods providing an efficient methodology for predicting cognitive development from longitudinal brain MRI data).