Signatures, Rough Paths and Applications in Machine Learning
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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.