Signature, Randomized Signature and Kernel Methods on Path Spaces.

This thesis explores the use of Signatures in Machine Learning through the lens of Kernel Methods. Signatures are central objects in the theory of Rough Paths which have found wide application in the Machine Learning domain promising to be canonical feature extractors on path spaces. Related Kernel Methods have recently received particular attention being easily computable using off-the-shelf PDE solvers. Randomized Signatures behave like signature but are much easier to compute being solutions of simple, random, and finite dimensional Controlled Differential Equations. They present promising results even though some aspects have yet to be rigorously studied. This work is divided in three main parts: 1 - We introduce the mathematics behind the paradigm of Kernel Learning. 2 - We frame Signatures in the Machine Learning context and analyze related Kernel Methods: Signature Kernels (SK). 3 - We try to rigorously define Randomized Signatures and prove central results. We then proceed to study the properties of the novel Randomized Signature Kernel (rSK) and end with the proof of a conjecture relating rSK and SK