Alexander Schell

Abstract

Nonparametric regression of stochastic processes, which shares conceptual foundations with large language models, enables the estimation of statistical relationships in multidimensional, time-dependent data without relying on fixed parametric assumptions. In this talk, we present a novel approach to this classical estimation problem using the signature transform from rough path theory. This transform encodes essential information about a stochastic process as a sequence of iterated integrals, capturing its statistical properties globally in time and hierarchically. By framing statistical regression as an operator learning problem, we show that this signature-based discretisation characterises the conditional dependence of one stochastic process on another as the solution to a convex, semi-infinite linear least squares problem. This characterisation, which relies on a functional monotone class argument involving the bounded signature of the conditioning process, leads to efficient and universally consistent nonparametric estimators for regression functions (~ conditional expectations) and conditional distributions across very general classes of jointly distributed stochastic processes. These estimators are practically computable as solutions to convex optimisation problems and are supported by broad theoretical guarantees. 

Our speaker

Alexander is a postdoctoral researcher at the Seminar for Applied Mathematics at ETH Zurich, where he works at the intersection of stochastic analysis and machine learning, with a focus on statistical inverse problems and inference from stochastic dynamical systems. Previously, he was a postdoc at the Department of Statistics at Columbia University and obtained his DPhil in Mathematics from the University of Oxford in 2022 under the supervision of Harald Oberhauser. Prior to that, Alexander completed an MSc in Pure Mathematics at Imperial College London and a BSc and MSc in Mathematics with a minor in Theoretical Physics at Ulm University, his hometown university in Germany.

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