Workshop on Stochastic Analysis, Geometry and Statistics

Overview

The theme of the SAGEST workshop is the use of geometric methods in Stochastic Analysis and Statistics, with a particular focus on the following topics:

Geometry and Stochastic Analysis

The UK is at the forefront of rough paths theory from the founding work of Terry Lyons at Oxford through to the recent Fields medal winning work of Martin Hairer which applied techniques from rough paths to give rigorous foundations to important stochastic partial differential equations of mathematical physics (the KPZ and 3-dimensional stochastic quantization equations). To understand these equations, important new theories have been developed: those of regularity structures and of paracontrolled distributions, the theory of controlled rough paths providing an important ingredient in both. In addition to the ones in mathematical physics, rough path theory has found applications in statistics, machine learning and finance. The workshop will bring together researchers interested both in the pure mathematical developments and in their applications.

Information Geometry

Mark Girolami and his co-authors proposed a Riemannian Manifold Hamiltonian Monte Carlo sampler to resolve the shortcomings of existing Monte Carlo algorithms when sampling from target densities that may be high dimensional and exhibit strong correlations. This approach exploits the Riemannian geometry of statistical manifolds to optimise scaling behaviour and results allow efficient sampling even in high dimensions. The study of Hamiltonian Monte Carlo is growing rapidly and has interactions with the study of stochastic analysis on manifolds. Stochastic processes on statistical manifolds have also been discussed by Brigo et al. in connection with dimensionality reduction of stochastic filtering schemes based on projections. The notion of projection for stochastic differential equations poses questions that are not appreciated in the classical setting of ordinary differential equations. Adopting the perspective of Rough Differential Equations to tackle such questions is an emerging area of research, currently being explored through collaborations among UK researchers such as D. Brigo, M. Girolami and G. Pavliotis, N. Newton and T. Cass.

Variational Approach to Fokker-Planck

Since the seminal paper by Jordan, Kinderlehrer and Otto, the Fokker-Planck equation dynamics has been regarded as a gradient flux, or a steepest descent, for free energy with respect to the Wasserstein metric. This has opened a new area in PDEs research and many results related also to optimal transport have been formulated. Potential connections with the two areas above are becoming clear among researchers, especially with optimal approximation of the Fokker-Planck equation and  with Hamiltonian Monte Carlo, leveraging on the common geometry techniques and metrics.

Organisers and participants

Organising committee

• John Armstrong (KCL)
• Damiano Brigo
• Thomas Cass

Conference secretariat

• Claudio Bellani
• Emilio Rossi Ferrucci

Funding bodies

• CFM – Imperial Institute of Quantitative Finance
• Department of Mathematics, Imperial College London
• Department of Mathematics, King’s College London
• London Mathematical Society

Invited speakers

• Michael Betancourt (Symplectomorphic, LLC)
• David Elworthy (Warwick)
• Peter Friz (TU Berlin)
• Andrew Duncan (Sussex)
• Karen Habermann (Bonn)
• Martin Hairer (Imperial College London)
• Yves Le Jan (Université Paris-Sud)
• Xue-Mei Li (Imperial College London)
• Terry Lyons (Oxford)
• Nigel Newton (Essex)
• James Norris (Cambridge)
• Grigorios Pavliotis (Imperial College London)
• Giovanni Pistone (Turin)
• Amanda Turner (Lancaster)

Conference Schedule

List of abstracts

Michael Betancourt
The Geometric Foundations of Hamiltonian Monte Carlo

Evaluating expectation values with respect to a complex probability distribution is a fundamental problem in computational statistics.  In particular, methodologies capable of parsimoniously exploring these distributions, and hence efficiently and accurately estimating expectation values, are essential in realizing effective computation in practice.  A powerful way of generating this efficient exploration is by exploiting a measure-preserving flow, that is if one can be found for the target distribution.  Fortunately Hamiltonian Monte Carlo provides a procedure for automatically constructing such flows for distributions on any smooth manifold.  In this talk I will discuss the differential geometric foundations of Hamiltonian Monte Carlo, from its formal construction to its optimal implementation and powerful diagnostic properties.

Andrew Duncan
The Information Geometry arising from Kernel-based Discrepancies

A major limitation of likelihood based inference methods is their inability to handle unnormalised and/or general generative models. This has motivated the introduction of estimators based on novel divergences, for example the Hyvarinen score. In this talk we discuss a family of divergences which characterise the discrepancy between two measures as the maximum error in expectation over a class of functions. When the underlying set of functions is taken to be the unit ball of an appropriately constructed reproducing kernel Hilbert space these divergences become tractable. Two examples are the Maximum Mean Discrepancy and Kernel Stein disrepancy. In this talk I will discuss minimal discrepancy estimators based on these divergences, focusing on the information geometry induced on the model family, its properties and its implications on the efficiency and robustness of such estimators. Secondly, I will describe our efforts to improve the efficiency of discrepancy minimisers by introducing an appropriate natural gradient.

David Elworthy
Higher derivatives of heat kernels: a geometric approach

I will describe how earlier work with Yves LeJan & Xue-Mei Li can be applied to give path integral formulae and estimates for higher derivatives of heat kernels on manifolds, demonstrating that the situation can be both more complicated and more interesting than might at first be expected.

Peter Friz
Varieties of Signature Tensors

The signature of a parametric curve is a sequence of tensors whose entries are iterated integrals. This construction is central to the theory of rough paths in stochastic analysis. It is here examined through the lens of algebraic geometry. We introduce varieties of signature tensors for both deterministic and random paths. For the former, we focus on piecewise linear paths, on polynomial paths, and on varieties derived from free nilpotent Lie groups. For the latter, we focus on Brownian motion and its mixtures. Joint work with Carlos Améndola and Bernd Sturmfels.

Karen Habermann
Small-time fluctuations for conditioned hypoelliptic diffusions

After giving an overview of work by Bailleul, Mesnager and Norris on the small-time fluctuations for sub-Riemannian diffusion bridges, which assumes the initial and final positions to lie outside the sub-Riemannian cut locus, we extend their results to the diagonal and describe the asymptotics of sub-Riemannian diffusion loops. We show that, in a suitable chart and after a suitable rescaling, the small-time diffusion loop measures have a non-degenerate limit, which we identify explicitly in terms of a certain local limit operator.

Both works are concerned with the point-to-point setting for hypoelliptic diffusion processes, whose associated generators satisfy the so-called strong Hörmander condition. We further present work in progress illustrating small-time phenomena which can occur when one instead only assumes a weak Hörmander condition and where one also considers the point-to-subspace setting.

Martin Hairer
Stochastic quantisation of Yang-Mills

Yves Le Jan
Markovian Networks

Markov loop ensembles on graphs are shown to be related to random maps, networks, and flows. Explicit distributions are computed. Finally, it is shown how some results can be extended to manifolds.

Xue-Mei Li
Group action and Horizontal lifts

Consider a stochastic dynamics invariant under a group action. Given a small perturbation, we discuss the effect of the perturbation to the orbit of the stochastic action and on how to use horizontal lift to extract an effective limit.

Terry Lyons
Title TBA

Nigel Newton
Nonparametric Information Geometry and Nonlinear Filtering

The talk will describe a class of infinite-dimensional (non-parametric) manifolds of probability measures, and discuss their application to Nonlinear Filtering.  The manifolds differ in the number of derivatives possessed by the densities of their members. Each admits a weak Riemannian metric (the Fisher-Rao metric) and a one-parameter family of affine connections (the Amari-Chentsov alpha-connections).  The manifolds are dually (+-1)-flat, but have non-zero metric curvature.  They are variously modelled on the Banach spaces, C^k_b(B;R), and the Fréchet space C^\infty_b(B;R), where B is an open subset of a Banach space.

Nonlinear Filtering is a branch of Bayesian estimation, in which a “signal process” is progressively estimated from the history of an “observations process”.  Nonlinear filters are often represented by stochastic differential equations for the posterior distribution of the signal, which is typically of infinite dimension in the sense that it cannot be represented by a finite number of statistics.  It will be argued that the statistical divergences of Information Geometry are natural measures of error for approximations to such distributions, and that non-parametric statistical manifolds are natural “state spaces” for filters.  The two “vector fields” of nonlinear filtering are (respectively) (+-1)-affine, and so dually  (+-1)-flat manifolds are particularly attractive.  The talk will also outline some information geometric properties of nonlinear filters, which are of interest in Nonequilibrium Statistical Mechanics.

James Norris
Scaling limits for planar aggregation with subcritical fluctuations

A two-dimensional cluster, growing by aggregation of a sequence of particles, may be encoded as a composition of conformal maps. This offers a means to formulate and analyse models for planar random growth. I will focus on scaling limits in the case where there are many small particles, first for the case where the conformal maps are chosen to be independent, and then for a variant model which takes the fluctuations of the process towards a critical point, which is a limit of stability. Joint work with Vittoria Silvestri and Amanda Turner.

Greg Pavliotis
Long-time behaviour and phase transitions for the McKean-Vlasov equation

We study the long time behaviour and the number and structure of stationary solutions for the McKean-Vlasov equation, a nonlinear nonlocal Fokker-Planck type equation that describes the mean field limit of a system of weakly interacting diffusions. We consider two cases: the McKean-Vlasov equation in a multiscale confining potential with quadratic, Curie-Weiss, interaction (the so-called Dasai-Zwanzig model), and the McKean-Vlasov dynamics on the torus with periodic boundary conditions and with a localized interaction. Our main objectives are the study of convergence to a stationary state and the construction of the bifurcation diagram for the stationary problem. The application of our work to the study of models for opinion formation is also discussed.

Giovanni Pistone
Exponential Manifold on the Gaussian Space: Orlicz-Sobolev Spaces, Fokker-Plank Equation, Hyvärinen Divergence, Wasserstein Metric

Given a Gaussian space (R^n, γ), we consider the set M of positive densities p that are connected to the unit density by an open Hellinger arc. The elements of M are precisely the densities of the form exp(u−K(u)) where E_γ(u) = 0, K(u) is a normalising constant, and u belongs to the exponential Orlicz space with weight γ i.e., E_γ(cosh(αu)) is finite for some α. M is the exponential manifold for an affine atlas of charts with values in the exponential Orlicz space with weight p. The general set-up is described in [Pistone 2013] and [Santacroce et al. 2016]. The Gaussian assumption provides the exponential manifold with special features. In particular, it allows to connect the geometry of densities with the geometry of the sample space, e.g. [Lods and Pistone 2015].

We discuss the manifold of smooth densities by taking as model space of the exponential manifold the Orlicz-Sobolev spaces with Gaussian weight γ. It is a research project in progress. The first version appears in [Pistone, 2018], where some preliminary results are discussed, Poincaré inequalities and translation statistical models. Statistical applications involving smooth densities will be discussed in particular, the finite-dimensional projection of evolution equations ([Nielsen, Critchley, Dodson 2017], [Brigo and Pistone 2017]). Other application of interest are Hyvärinen divergence [Hyvärinen 2005] and the Wasserstein geometry of M [Otto 2001].

Amanda Turner
Scaling limits of planar random growth processes

Planar random growth processes occur widely in the physical world. Examples include diffusion-limited aggregation (DLA) for mineral deposition and the Eden model for biological cell growth. One approach to mathematically modelling such processes is to represent the randomly growing clusters as compositions of conformal mappings. In 1998, Hastings and Levitov proposed one such family of models, which includes versions of the physical processes described above. In this talk, I will discuss recent progress in obtaining scaling limits for a natural generalisation of the Hastings-Levitov family.