University of Oxford and Imperial College London

2:30 -3.00 Will Turner (CDT Student, Imperial College London)

Topologies on unparameterised path space

The signature of a path is a non-commutative exponential introduced by K.T. Chen in the 1950s, and appears as a central object in the theory of rough paths developed by T. Lyons in the 1990s. For continuous paths of bounded variation, the signature may be realised as a sequence of iterated integrals, which provides a succinct summary for multimodal, irregularly sampled, time-ordered data. The terms in the signature act as an analogue to monomials for finite dimensional data: linear functionals on the signature uniformly approximate any compactly supported continuous function on unparameterised path space (Levin, Lyons, Ni 2013). Selection of a suitable topology on the space of unparameterised paths is then key to the practical use of this approximation theory. We present new results on the properties of several candidate topologies for this space. If time permits, we will relate these results to two classical models: the fixed-time solution of a controlled differential equation, and the expected signature model of Levin, Lyons, and Ni. This is joint work with Thomas Cass.

3.05 -3.35 Ross Zhang (CDT Student, University of Oxford)

Random vortex dynamics via functional stochastic differential equations

The talk focuses on the representation of the three-dimensional (3D) Navier-Stokes equations by a random vortex system. This new system could give us new numerical schemes to efficiently approximate the 3D incompressible fluid flows by Monte Carlo simulations. Compared with the 2D Navier-Stokes equation, the difficulty of the 3D Navier-Stokes equation lies in the stretching of vorticity. To handle the stretching term, a system of stochastic differential equations is coupled with a functional ordinary differential equation in the 3D random vortex system. Two main tools are developed to derive the new system: the first is the investigation of pinned diffusion measure, which describes the conditional distribution of a time reversal diffusion, and the second is a forward-type Feynman Kac formula for nonlinear PDEs, which utilizes the pinned diffusion measure to delicately overcome the time reversal issue in PDE. Although the main focus of the research is the Navier-stokes equation, the tools developed in this research are quite general. They could be applied to other nonlinear PDEs as well, thereby providing respective numerical schemes.

3.40 - 4.25pm Dr Cris Salvi (Imperial College London)

Signature kernel methods

Kernel methods provide a rich and elegant framework for a variety of learning tasks including supervised learning, hypothesis testing, Bayesian inference, generative modelling and scientific computing. Sequentially ordered information often arrives in the form of complex streams taking values in non-trivial ambient spaces (e.g. a video is a sequence of images). In these situations, the design of appropriate kernels is a notably challenging task. In this talk, I will outline how rough path theory, a modern mathematical framework for describing complex evolving systems, allows to construct a family of characteristic kernels on pathspace known as signature kernels. I will then present how signature kernels can be used to develop a variety of algorithms such as two-sample hypothesis and (conditional) independence tests for stochastic processes, generative models for time series and numerical methods for path-dependent PDEs.

4.30 Refreshments

1pm Jonathan Tam: Markov decision processes with observation costs

We present a framework for a controlled Markov chain where the state of the chain is only given at chosen observation times and of a cost. Optimal strategies therefore involve the choice of observation times as well as the subsequent control values. We show that the corresponding value function satisfies a dynamic programming principle, which leads to a system of quasi-variational inequalities (QVIs). Next, we give an extension where the model parameters are not known a priori but are inferred from the costly observations by Bayesian updates. We then prove a comparison principle for a larger class of QVIs, which implies uniqueness of solutions to our proposed problem. We utilise penalty methods to obtain arbitrarily accurate solutions. Finally, we perform numerical experiments on three applications which illustrate our framework.

Preprint at https://arxiv.org/abs/2201.07908

1.45pm Remy Messadene: signature asymptotics, empirical processes, and optimal transport

Rough path theory provides one with the notion of signature, a graded family of tensors which characterise, up to a negligible equivalence class, and ordered stream of vector-valued data. In the last few years, use of the signature has gained traction in time-series analysis, machine learning, deep learning and more recently in kernel methods. In this work, we lay down the theoretical foundations for a connection between signature asymptotics, the theory of empirical processes, and Wasserstein distances, opening up the landscape and toolkit of the second and third in the study of the first. Our main contribution is to show that the Hambly-Lyons limit can be reinterpreted as a statement about the asymptotic behaviour of Wasserstein distances between two independent empirical measures of samples from the same underlying distribution. In the setting studied here, these measures are derived from samples from a probability distribution which is determined by geometrical properties of the underlying path.

2.30-3.00 Tea & coffee in the mezzananie

3-4pm Julien Berestycki: Extremal point process of the branching Brownian motion