Oxford

Many computational techniques in data science involve the factorization of a data matrix into the product of two or more structured matrices. Examples include PCA, which relies on computing an SVD, recommendation systems, which leverage non-negative matrix factorization, infilling missing entries with low rank matrix completion, and finding sparse representations via dictionary learning. In our work we study a new matrix factorization problem, involving the recovery of $\textbf{A}$ and $\textbf{X}$ from $\textbf{Y} := \textbf{A}\textbf{X}$ under the following assumptions; $\textbf{A}$ is an $m \times n$ sparse binary matrix with a fixed number $d$ of nonzeros per column and $\textbf{X}$ is an $n \times N$ sparse real matrix whose columns have $k$ nonzeros and are dissociated. This setup is inspired and motivated by similar models studied in the dictionary learning literature as well as potential connections both with stochastic block models and combinatorial compressed sensing. In this talk we present a new algorithm, EBR, for solving this problem, as well as recovery guarantees in the context of a particular probabilistic data model. Using the properties of expander graphs we are able to show, under certain assumptions, that with just $N = \textit{O}( \log^2(n))$ samples then EBR recovers the factorization up to permutation with high probability. 

We extend the lightning Laplace solver (Gopal and Trefethen, SINUM 2019) to unbounded domains and to the biharmonic equation. Illustrating the high accuracy of such methods, we get beautiful contour plots of Moffatt eddies.

We show that the scalability challenges of Global Optimisation algorithms can be overcome for functions with low effective dimensionality, which are constant along certain linear subspaces. Such functions can often be found in applications, for example, in hyper-parameter optimization for neural networks, heuristic algorithms for combinatorial optimization problems and complex engineering simulations. We propose the use of random subspace embeddings within a(ny) global minimisation algorithm, extending the approach in Wang et al. (2016). Using tools from random matrix theory and conic integral geometry, we investigate the efficacy and convergence of our random subspace embeddings approach, in a static and/or adaptive formulation. We illustrate our algorithmic proposals and theoretical findings numerically, using state of the art global solvers. This work is joint with Coralia Cartis.
 

Subspace correction (SSC) methods are based on a "divide and conquer" strategy, where a global problem is divided into a sequence of local ones. This framework includes iterative methods such as Jacobi iteration, domain decomposition, and multigrid methods. We are interested in nonlinear PDEs exhibiting highly localized nonlinearities, for which Newton's method can take many iterations. Instead of doing all this global work, nonlinear SSC methods tackle the localized nonlinearities within subproblems. In this presentation, we describe the SSC framework, and investigate combining Newton's method with nonlinear SSC methods to solve a regularized Stefan problem.
 

Vandermonde matrices are exponentially ill-conditioned, rendering the familiar “polyval(polyfit)” algorithm for polynomial interpolation and least-squares fitting ineffective at higher degrees. We show that Arnoldi orthogonalization fixes the problem.

I will discuss the geometric interpretation of the twisted index of 3d supersymmetric gauge theories on a closed Riemann surface. In the first part of the talk, I will show that the twisted index computes the virtual Euler characteristic of the moduli space of solutions to vortex equations on the Riemann surface, which can be understood algebraically as quasi-maps to the Higgs branch. I will explain 3d N=4 mirror symmetry in this context, which implies non-trivial relations between enumerative invariants associated to these moduli spaces. Finally, I will present a wall-crossing formula for these invariants derived from the gauge theory point of view.
 

This talk will explain how Lie polynomials underpin the structure of the so-called double copy relationship between gauge and gravity theories (and a network of other theories besides).  ABHY have recently shown that Lie polynomials arise naturally also in the geometry of the space K_n of momentum invariants, Mandelstams, and can be expressed in the space of n-3-forms dual to certain associahedral (n-3)-planes. They also arise in the moduli space M_{0,n} of n points on a Riemann sphere up to Mobius transformations in the n-3-dimensional homology.  The talk goes on to give a natural correspondendence between K_n and the cotangent bundle of M_{0.n} through which the relationships of some of these structures can be expressed.  This in particular gives a natural framework for expressing the CHY and ambitwistor-string formulae for scattering amplitudes of gauge and gravity theories and goes some way to expressing their double copy relations.   This is part of joint work in progress with Hadleigh Frost.

The principal ideal theorem (1930) guaranteed that any number field K would embed into a finite extension, called the Hilbert class field of K, in which every ideal of the original field became principal -- however the Hilbert class field itself will not necessarily have class number 1. The class field tower problem asked whether iteratively taking Hilbert class fields must stabilize after finitely many steps. In 1964, it was finally answered in the negative by Golod and Shafarevich who produced infinitely many examples and pioneered the framework that is still the most common setting for deciding when a number field will have an infinite class field tower.

In this talk, I will finish the proof of their cohomological result and thus fully justify how it settled the class field tower problem.

The principal ideal theorem (1930) ascertained that any number field K embeds into a finite extension, called the Hilbert class field of K, in which every ideal of the original field became principal -- however the Hilbert class field itself will not necessarily have class number 1. The class field tower problem asked whether iteratively taking Hilbert class fields must stabilize after finitely many steps. In 1964, it was finally answered in the negative by Golod and Shafarevich who produced infinitely many examples and pioneered the framework that is still the most common setting for deciding when a number field will have an infinite class field tower.

In this talk, I will sketch the proof of their cohomological result and explain how it settled the class field tower problem.