Mathematical Institute

Identifying discrete patterns in binary data is an important dimensionality reduction tool in machine learning and data mining. In this paper, we consider the problem of low-rank binary matrix factorisation (BMF) under Boolean arithmetic. Due to the NP-hardness of this problem, most previous attempts rely on heuristic techniques. We formulate the problem as a mixed integer linear program and use a large scale optimisation technique of column generation to solve it without the need of heuristic pattern mining. Our approach focuses on accuracy and on the provision of optimality guarantees. Experimental results on real world datasets demonstrate that our proposed method is effective at producing highly accurate factorisations and improves on the previously available best known results for 16 out of 24 problem instances.

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In this talk, we will present an approach to constrained optimisation when the set of constraints is a smooth manifold. This setting is of particular interest in data science applications, as many interesting sets of matrices have a manifold structure. We will show how we may couple classic ideas from differential geometry with modern methods such as autodifferentiation to simplify optimisation problems from spaces with a difficult topology (e.g. problems with orthogonal or fixed-rank constraints) to problems on ℝⁿ where we can use any classical optimisation methods to solve them. We will also show how to use these methods to automatically compute quantities such as the Riemannian gradient and Hessian. We will present the library GeoTorch that allows for putting these kind of constraints within models written in PyTorch by adding just one line to the model. We will also comment on some convergence results if time allows.

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Preconditioning techniques are widely used for speeding up the iterative solution of systems of linear equations, often by transforming the system into one with lower condition number. Even though the condition number also serves as the determining constant in simple bounds for the numerical error of the solution, simple experiments and bounds show that such preconditioning on the matrix level is not guaranteed to reduce this error. Transformations on the operator level, on the other hand, improve both accuracy and speed of iterative methods as predicted by the change of the condition number. We propose to investigate such methods under a common framework, which we call full operator preconditioning, and show practical examples.

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This week Professor Dominic Vella will talk about wrinkling  

In this talk I will provide an overview of recent work on the wrinkling of thin elastic objects. In particular, the focus of the talk will be on answering questions that arise in recent applications that seek not to avoid, but rather, exploit wrinkling. Such applications usually take place far beyond the threshold of instability and so key themes will be the limitations of “standard” instability analysis, as well as what analysis should be performed instead. I will discuss the essential ingredients of this ‘Far-from-Threshold’ analysis, as well as outlining some open questions.  

Abstract

This talk will provide an overview of mathematical modelling applied to the behaviour of ice sheets and their role in the climate system.  I’ll provide some motivation and background, describe simple approaches to modelling the evolution of the ice sheets as a fluid-flow problem, and discuss some particular aspects of the problem that are active areas of current research.  The talk will involve a variety of interesting continuum-mechanical models and approximations that have analogues in other areas of applied mathematics.

You can join the meeting by clicking on the link below.
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The many facets of community detection on networks 

Community detection, the decomposition of a graph into essential building blocks, has been a core research topic in network science over the past years. Since a precise notion of what consti- tutes a community has remained evasive, community detection algorithms have often been com- pared on benchmark graphs with a particular form of assortative community structure and classified based on the mathematical techniques they employ. However, this comparison can be misleading because apparent similarities in their mathematical machinery can disguise different goals and rea- sons for why we want to employ community detection in the first place. Here we provide a focused review of these different motivations that underpin community detection. This problem-driven classification is useful in applied network science, where it is important to select an appropriate algorithm for the given purpose. Moreover, highlighting the different facets of community detection also delineates the many lines of research and points out open directions and avenues for future research.