Signatures, Rough Paths and Applications in Machine Learning
Summary
In the context of rough paths theory, the signature is a fundamental object that
captures information about paths. Recent developments in this area have motivated
the use of signatures as a nonparametric feature set for machine learning.
With this thesis, we aim to provide an accessible introduction to signatures from
both a theoretical and applied point of view. On the theoretical side, we review
algebraic, analytic and geometric properties of the signature. This touches on many
different topics in pure mathematics, including differential equations, multilinear
algebra and Lie theory. On the applied side, we review how these properties
enable signatures to be effective feature sets in supervised learning problems. We
study general algorithms and implementations for computing signatures, as well as
specialized transformations for machine learning. Several classical problems such
as handwritten digit recognition are explored in detail. Our main contribution to
the existing literature is a benchmark of signatures for time series classification
problems from the UCR repository.