University of Edinburgh

Deep generative models provide powerful tools for fitting difficult distributions such as modelling natural images. But many of these methods, including  variational autoencoders (VAEs) and generative adversarial networks (GANs), can be notoriously difficult to fit.

One well-known problem is mode collapse, which means that models can learn to characterize only a few modes of the true distribution. To address this, we introduce VEEGAN, which features a reconstructor network, reversing the action of the generator by mapping from data to noise. Our training objective retains the original asymptotic consistency guarantee of GANs, and can be interpreted as a novel autoencoder loss over the noise.

Second, maximum mean discrepancy networks (MMD-nets) avoid some of the pathologies of GANs, but have not been able to match their performance. We present a new method of training MMD-nets, based on mapping the data into a lower dimensional space, in which MMD training can be more effective. We call these networks Ratio-based MMD Nets, and show that somewhat mysteriously, they have dramatically better performance than MMD nets.

A final problem is deciding how many latent components are necessary for a deep generative model to fit a certain data set. We present a nonparametric Bayesian approach to this problem, based on defining a (potentially) infinitely wide deep generative model. Fitting this model is possible by combining variational inference with a Monte Carlo method from statistical physics called Russian roulette sampling. Perhaps surprisingly, we find that this modification helps with the mode collapse problem as well.

We discuss two Freidlin-Wentzell large deviation principles for McKean-Vlasov equations (MV-SDEs) in certain path space topologies. The equations have a drift of polynomial growth and an existence/uniqueness result is provided. We apply the Monte-Carlo methods for evaluating expectations of functionals of solutions to MV-SDE with drifts of super-linear growth.  We assume that the MV-SDE is approximated in the standard manner by means of an interacting particle system and propose two importance sampling (IS) techniques to reduce the variance of the resulting Monte Carlo estimator. In the "complete measure change" approach, the IS measure change is applied simultaneously in the coefficients and in the expectation to be evaluated. In the "decoupling" approach we first estimate the law of the solution in a first set of simulations without measure change and then perform a second set of simulations under the importance sampling measure using the approximate solution law computed in the first step. 

Archimedes, who famously jumped out of his bath shouting "Eureka", also invented $\pi$. 

Euler invented $e$ and had fun with his formula $e^{2\pi i} = 1$

The world is full of important numbers waiting to be invented. Why not have a go ?

Michael Atiyah is one of the world's foremost mathematicians and a pivotal figure in twentieth and twenty-first century mathematics. His lecture will be followed by an interview with Sir John Ball, Sedleian Professor of Natural Philosophy here in Oxford where Michael will talk about his lecture, his work and his life as a mathematician.

Please email @email to register.

The Oxford Mathematics Public Lectures are generously supported by XTX Markets.

Many systems in nature consist of stochastically interacting agents or particles. Stochastic processes have been widely used to model such systems, yet they are notoriously difficult to analyse. In this talk I will show how ideas from statistics can be used to tackle some challenging problems in the field of stochastic processes.

In the first part, I will consider the problem of inference from experimental data for stochastic reaction-diffusion processes. I will show that multi-time distributions of such processes can be approximated by spatio-temporal Cox processes, a well-studied class of models from computational statistics. The resulting approximation allows us to naturally define an approximate likelihood, which can be efficiently optimised with respect to the kinetic parameters of the model. 

In the second part, we consider more general path properties of a certain class of stochastic processes. Specifically, we consider the problem of computing first-passage times for Markov jump processes, which are used to describe systems where the spatial locations of particles can be ignored.  I will show that this important class of generally intractable problems can be exactly recast in terms of a Bayesian inference problem by introducing auxiliary observations. This leads us to derive an efficient approximation scheme to compute first-passage time distributions by solving a small, closed set of ordinary differential equations.

In the ongoing programme to classify noncommutative projective surfaces (connected graded noetherian domains of Gelfand-Kirillov dimension three) a natural question is:  what are the minimal models within a birational class?  It is not even clear a priori what the correct definition is of a minimal model in this context.

We show that a generic Sklyanin algebra (a noncommutative analogue of P^2) satisfies the surprising property that it has no birational connected graded noetherian overrings, and explain why this is a reasonable definition of 'minimal model.' We show also that the noncommutative versions of P^1xP^1 and of the Hirzebruch surface F_2 are minimal.
This is joint work in progress with Dan Rogalski and Toby Stafford.

 

The talk will be based on some of the material in the joint survey with Etienne Ghys

"Signatures in algebra, topology and dynamics"

http://arxiv.org/abs/1512.092582

In the 19th century Sturm's theorem on the number of roots of a real polynomial motivated Sylvester to define the signature of a quadratic form. In the 20th century the classification of quadratic forms over algebraic number fields motivated Witt to introduce the "Witt groups" of stable isomorphism classes of quadratic forms over arbitrary fields. Still in the 20th century the study of high-dimensional topological manifolds with nontrivial fundamental group motivated Wall to introduce the "Wall groups" of stable isomorphism classes of quadratic forms over arbitrary rings with involution. In our survey we interpreted Sturm's theorem in terms of the Witt-Wall groups of function fields. The talk will emphasize the common thread running through this developments, namely the notion of the localization of a ring inverting elements. More recently, the Cohn localization of inverting matrices over a noncommutative ring has been applied to topology in the 21st century, in the context of the speaker's algebraic theory of surgery.

Leinster and Willerton have introduced the concept of the magnitude of a metric space, as a special case as that of an enriched category. It is a numerical invariant which is designed to capture the important geometric information about the space, but concrete examples of ts values on compact sets in euclidean space have hitherto been lacking. We discuss progress in some conjectures of Leinster and Willerton.

The concept of delegated quantum computing is a quantum extension of  
the classical task of computing with encrypted data without decrypting  
them first. Many quantum protocols address this challenge for a  
futuristic quantum client-server setting achieving a wide range of  
security properties. The central challenge of all these protocols to  
be applicable for classical tasks (such as secure multi party  
computation or fully homomorphic encryption) is the requirement of a  
server with a universal quantum computer. By restricting the task to  
classical computation only, we derive a protocol for unconditionally  
secure delegation of classical computation to a remote server that has  
access to basic quantum devices.

Security Constrained Optimal Power Flow is an increasingly important problem for power systems operation both in its own right and as a subproblem for more complex problems such as transmission switching or
unit commitment.

The structure of the problem resembles stochastic programming problems in that one aims to find a cost optimal operation schedule that is feasible for all possible equipment outage scenarios
(contingencies). Due to the presence of power flow constraints (in their "DC" or "AC" version), the resulting problem is a large scale linear or nonlinear programming problem.

However it is known that only a small subset of the contingencies is active at the solution. We show how Interior Point methods can exploit this structure both by simplifying the linear algebra operations as
well as generating necessary contingencies on the fly and integrating them into the algorithm using IPM warmstarting techniques. The final problem solved by this scheme is significantly smaller than the full
contingency constrained problem, resulting in substantial speed gains.

Numerical and theoretical results of our algorithm will be presented.