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Randomized Signatures and Applications in Machine Learning

This thesis explores the intersection of rough path and signature theory, reservoir computing, and machine learning, focusing on the concept of the randomized signature. The randomized signature is a dimensionality-reduced path signature that is obtained by random projection from the original signature space to a lower dimensional vector space. The projections maps, as well as the error bounds of this approximation are derived from the classical Johnson-Lindenstrauss lemma. It is shown that the randomized signature retains the favorable approximation properties of the original signature at a fraction of the computation cost, making it very attractive for applications in machine learning. The thesis provides a comprehensive theoretical foundation, touching upon controlled differential equations, rough path theory, and signature theory, as well as the connections to reservoir computing and machine learning, before delving into definitions and proofs of the randomized signature. It showcases the computational advantages and approximation capabilities of the randomized signature compared to its original counterpart. Numerical experiments validate the efficacy and efficiency of randomized signature algorithms across various tasks, illustrating their utility in machine learning and quantitative finance.

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