Date: Wednesday 11 December 2PM - Room 210, Mathematical Sciences Building
Abstract: Scaling limits of random developments of a path into a matrix Lie group have recently been used to construct signature-based time series kernels. General linear group developments have been shown to be connected to the ordinary signature kernel (Muça Cirone et al.), while unitary developments have been used to construct the path characteristic function distance (Lou et al.) which has proven a successful discriminator for generative modelling tasks. By leveraging the tools of random matrix theory and free probability theory, we are able to provide a unified treatment of both limits under general assumptions on the randomisation. 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. Joint work with Thomas Cass.