Oxford

The talk introduces the problem of completing a partially observed matrix whose columns obey a nonlinear structure. This is an extension of classical low-rank matrix completion where the structure is linear. Such matrices are in general full rank, but it is often possible to exhibit a low rank structure when the data is lifted to a higher dimensional space of features. The presence of a nonlinear lifting makes it impossible to write the problem using common low-rank matrix completion formulations. We investigate formulations as a nonconvex optimisation problem and optimisation on Riemannian manifolds.

This work determines an aim point selection strategy for players in order to improve their chances of winning at the classic darts game of 501. Although many previous studies have considered the problem of aim point selection in order to maximise the expected score a player can achieve, few have considered the more general strategical question of minimising the expected number of turns required for a player to finish. By casting the problem as a Markov decision process, a framework is derived for the identification of the optimal aim point for a player in an arbitrary game scenario.

There is no more classical problem of numerical PDE than the Laplace equation in a polygon, but Abi Gopal and I think we are on to a big step forward. The traditional approaches would be finite elements, giving a 2D representation of the solution, or integral equations, giving a 1D representation. The new approach, inspired by an approximation theory result of Donald Newman in 1964, leads to a "0D representation" -- the solution is the real part of a rational function with poles clustered exponentially near the corners of the polygon. The speed and accuracy of this approach are remarkable. For typical polygons of up to 8 vertices, we can solve the problem in less than a second on a laptop and evaluate the result in a few microseconds per point, with 6-digit accuracy all the way up to the corner singularities. We don't think existing methods come close to such performance. Next step: Helmholtz?
 

Matrix completion is an area of great mathematical interest and has numerous applications, including recommender systems for e-commerce. The recommender problem can be viewed as follows: given a database where rows are users and and columns are products, with entries indicating user preferences, fill in the entries so as to be able to recommend new products based on the preferences of other users. Viewing the interactions between user and product instead as interactions between potential drug chemicals and disease-causing target proteins, the problem is that faced within the realm of drug discovery. We propose a divide and conquer algorithm inspired by the work of [1], who use recursive rank-1 approximation. We make the case for using an LP rank-1 approximation, similar to that of [2] by a showing that it guarantees a 2-approximation to the optimal, even in the case of missing data. We explore our algorithm's performance for different test cases.

[1]  Shen, B.H., Ji, S. and Ye, J., 2009, June. Mining discrete patterns via binary matrix factorization. In Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining (pp. 757-766). ACM.

[2] Koyutürk, M. and Grama, A., 2003, August. PROXIMUS: a framework for analyzing very high dimensional discrete-attributed datasets. In Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining (pp. 147-156). ACM.

Combining work of Galkin, Christopherson-Ilten, and Coates-Corti-Galkin-Golyshev-Kasprzyk we see that all smooth Fano threefolds admit a toric degeneration. We can use this fact to uniformly construct all Fano threefolds: given a choice of a fan we classify reflexive polytopes which break into unimodular pieces along this fan. We can then construct closed torus invariant embeddings of the corresponding toric variety using a technique - Laurent inversion - developed with Coates and Kaspzryk. The corresponding binomial ideal is controlled by the chosen fan, and in low enough codimension we can explicitly test deformations of this toric ideal. We relate the constructions we obtain to known constructions. We study the simplest case of the above construction, closely related to work of Abouzaid-Auroux-Katzarkov, in arbitrary dimension and use it to produce a tropical interpretation of the mirror superpotential via broken lines. We expect the computation to be the tropical analogue of a Floer theory calculation.

 In this seminar I will discuss a recently-found class of RG flows in four dimensions exhibiting enhancement of supersymmetry in the infrared, which provides a lagrangian description of several strongly-coupled N=2 SCFTs. The procedure involves starting from a N=2 SCFT, coupling a chiral multiplet in the adjoint representation of the global symmetry to the moment map of the SCFT and turning on a nilpotent expectation value for this chiral. We show that, combining considerations based on 't Hooft anomaly matching and basic results about the N=2 superconformal algebra, it is possible to understand in detail the mechanism underlying this phenomenon and formulate a simple criterion for supersymmetry enhancement. 

In the first part of this talk, I will present an extension of Chebfun, called Ballfun, for computing with functions and vectors in the unit ball. I will then describe an algorithm for solving the incompressible Navier-Stokes equations in the ball. Contrary to projection methods, we use the poloidal-toroidal decomposition to decouple the PDEs and solve scalars equations. The solver has an optimal complexity (up to polylogarithmic terms) in terms of the degrees of freedom required to represent the solution.

This talk features a self-contained introduction to persistent homology, which is the main ingredient of topological data analysis. 

This work deals with exploiting the low-dimensional hierarchical structure of speech signals towards the  goal  of  improving  acoustic  modelling using deep neural networks (DNN).  To this aim the work employ tools from sparsity aware signal processing under novel frameworks to enrich  the  acoustic  information  present  in  DNN posterior features. 

In the classical theory of fluid mechanics, a linear relationship between the stress and rate of strain is often assumed. Even when this relationship is non-linear, it is typically formulated in terms of an explicit relation. Implicit constitutive theories provide a theoretical framework that generalises this, allowing a, possibly multi-valued, implicit constitutive relation. Since it is not possible to solve explicitly for the stress in the constitutive relation, a more natural approach would be to include the stress as a fundamental unknown in the formulation of the problem. In this talk I will present a formulation with this feature and a proof of convergence of the finite element approximations to a solution of the original problem.