Imperial College London

Modern Data Assimilation (DA) can be traced back to the sixties and owes a lot to earlier developments in linear filtering theory. Since then, DA has evolved independently of Filtering Theory. To-date it is a massively important area of research due to its many applications in meteorology, ocean prediction, hydrology, oil reservoir exploration, etc. The field has been largely driven by practitioners, however in recent years an increasing body of theoretical work has been devoted to it. In this talk, In my talk, I will advocate the interpretation of DA through the language of stochastic filtering. This interpretation allows us to make use of advanced particle filters to produce rigorously validated DA methodologies. I will present a particle filter that incorporates three additional add-on procedures: nudging, tempering and jittering. The particle filter is tested on a two-layer quasi-geostrophic model with O(10^6) degrees of freedom out of which only a minute fraction are noisily observed.

Given a variety defined over a field of characteristic zero and an algebraically integrable foliation of corank less than or equal to two, we show the existence of a categorical quotient, defined on the non-empty open subset of algebraically smooth points, through which every invariant morphism factors uniquely. Some applications to quotients by connected groups will be discussed.
 

Persistent homology is an important methodology from topological data analysis which adapts theory from algebraic topology to data settings and has been successfully implemented in many applications. It produces a statistical summary in the form of a persistence diagram, which captures the shape and size of the data. Despite its widespread use, persistent homology is simply impossible to implement when a dataset is very large. In this talk, I will address the problem of finding a representative persistence diagram for prohibitively large datasets. We adapt the classical statistical method of bootstrapping, namely, drawing and studying smaller multiple subsamples from the large dataset. We show that the mean of the persistence diagrams of subsamples—taken as a mean persistence measure computed from the subsamples—is a valid approximation of the true persistent homology of the larger dataset. We give the rate of convergence of the mean persistence diagram to the true persistence diagram in terms of the number of subsamples and size of each subsample. Given the complex algebraic and geometric nature of persistent homology, we adapt the convexity and stability properties in the space of persistence diagrams together with random set theory to achieve our theoretical results for the general setting of point cloud data. We demonstrate our approach on simulated and real data, including an application of shape clustering on complex large-scale point cloud data.

This is joint work with Yueqi Cao (Imperial College London).

We consider the problem of parameter estimation for a McKean stochastic differential equation, and the associated system of weakly interacting particles. The problem is motivated by many applications in areas such as neuroscience, social sciences (opinion dynamics, cooperative behaviours), financial mathematics, statistical physics. We will first survey some model properties related to propagation of chaos and ergodicity and then move on to discuss the problem of parameter estimation both in offline and on-line settings. In the on-line case, we propose an online estimator, which evolves according to a continuous-time stochastic gradient descent algorithm on the asymptotic log-likelihood of the interacting particle system. The talk will present our convergence results and then show some numerical results for two examples, a linear mean field model and a stochastic opinion dynamics model. This is joint work with Louis Sharrock, Panos Parpas and Greg Pavliotis. Preprint: https://arxiv.org/abs/2106.13751

Kernel-based statistical algorithms have found wide success in statistical machine learning in the past ten years as a non-parametric, easily computable engine for reasoning with probability measures. The main idea is to use a kernel to facilitate a mapping of probability measures, the objects of interest, into well-behaved spaces where calculations can be carried out. This methodology has found wide application, for example two-sample testing, independence testing, goodness-of-fit testing, parameter inference and MCMC thinning. Most theoretical investigations and practical applications have focused on Euclidean data. This talk will outline work that adapts the kernel-based methodology to data in an arbitrary Hilbert space which then opens the door to applications for functional data, where a single data sample is a discretely observed function, for example time series or random surfaces. Such data is becoming increasingly more prominent within the statistical community and in machine learning. Emphasis shall be given to the two-sample and goodness-of-fit testing problems.

Let $G$ be a group acting transitively on a finite set $\Omega$. Then $G$ acts on $\Omega \times \Omega$ componentwise. Define the orbitals to be the orbits of $G$ on $\Omega \times \Omega$. The diagonal orbital is the orbital of the form $\Delta = \{(\alpha, \alpha) \mid \alpha \in \Omega \}$. The others are called non-diagonal orbitals. Let $\Gamma$ be a non-diagonal orbital. Define an orbital graph to be the non-directed graph with vertex set $\Omega$ and edge set $(\alpha,\beta) \in \Gamma$ with $\alpha, \beta \in \Omega$. If the action of $G$ on $\Omega$ is primitive, then all non-diagonal orbital graphs are connected. The orbital diameter of a primitive permutation group is the supremum of the diameters of its non-diagonal orbital graphs.

There has been a lot of interest in finding bounds on the orbital diameter of primitive permutation groups. In my talk I will outline some important background information and the progress made towards finding specific bounds on the orbital diameter. In particular, I will discuss some results on the orbital diameter of the groups of simple diagonal type and their connection to the covering number of finite simple groups. I will also discuss some results for affine groups, which provides a nice connection to the representation theory of quasisimple groups. 

Networks are a widely-used tool to investigate the large-scale connectivity structure in complex systems and graphons have been proposed as an infinite size limit of dense networks. The detection of communities or other meso-scale structures is a prominent topic in network science as it allows the identification of functional building blocks in complex systems. When such building blocks may be present in graphons is an open question. In this paper, we define a graphon-modularity and demonstrate that it can be maximised to detect communities in graphons. We then investigate specific synthetic graphons and show that they may show a wide range of different community structures. We also reformulate the graphon-modularity maximisation as a continuous optimisation problem and so prove the optimal community structure or lack thereof for some graphons, something that is usually not possible for networks. Furthermore, we demonstrate that estimating a graphon from network data as an intermediate step can improve the detection of communities, in comparison with exclusively maximising the modularity of the network. While the choice of graphon-estimator may strongly influence the accord between the community structure of a network and its estimated graphon, we find that there is a substantial overlap if an appropriate estimator is used. Our study demonstrates that community detection for graphons is possible and may serve as a privacy-preserving way to cluster network data.

arXiv link: https://arxiv.org/abs/2101.00503

Abstract: We formulate and solve a multi-player stochastic differential game between financial agents who seek to cost-efficiently liquidate their position in a risky asset in the presence of jointly aggregated transient price impact on the risky asset's execution price along with taking into account a common general price predicting signal. In contrast to an interaction of the agents through purely permanent price impact as it is typically considered in the literature on multi-player price impact games, accrued transient price impact does not persist but decays over time. The unique Nash-equilibrium strategies reveal how each agent's liquidation policy adjusts the predictive trading signal for the accumulated transient price distortion induced by all other agents' price impact; and thus unfolds a direct and natural link in equilibrium between the trading signal and the agents' trading activity. We also formulate and solve the corresponding mean field game in the limit of infinitely many agents and show how the latter provides an approximate Nash-equilibrium for the finite-player game. Specifically we prove the convergence of the N-players game optimal strategy to the optimal strategy of the mean field game.     (Joint work with Moritz Voss)
 

We consider the homogenization of a Stokes system in a domain having many small random holes. This model mainly arises from problems of solid-fluid interaction (e.g. the flow of a viscous and incompressible fluid through a porous medium). We aim at the rigorous derivation of the homogenization limit both in the Brinkmann regime and in the one of Darcy’s law. In particular, we focus on holes that are distributed according to probability measures that allow for overlapping and clustering phenomena.