Free probability, path developments and signature kernels as universal scaling limits

Abstract: Scaling limits of random developments of a path into a matrix Lie Group have recently been used to construct signature-based kernels on path space, while mitigating some of the dimensionality challenges that come with using signatures directly. Muça Cirone et al. have established a connection between the scaling limit of general linear group developments with Gaussian vector fields and the ordinary signature kernel, while Lou et al. utilised unitary group developments and previous work of Chevyrev and Lyons to construct a path characteristic function distance. By leveraging the tools of random matrix theory and free probability theory, we are able to provide a unified treatment of the limits in both settings under general assumptions on the vector fields. For unitary developments, we show that the limiting kernel is given by the contraction of a signature against the monomials of freely independent semicircular random variables. Using the Schwinger-Dyson equations, we show that this kernel can be obtained by solving a novel quadratic functional equation.

machine learningcommutative algebraalgebraic geometryalgebraic topologycombinatoricscategory theoryoperator algebrasrings and algebrasrepresentation theory

Audience: researchers in the topic

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Algebraic and Combinatorial Perspectives in the Mathematical Sciences

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