# Discrete Mathematics and Data Science Research Team

The Discrete Mathematics and Data Science (DM&DS) interdisciplinary team considers discrete and data related interdisciplinary research topics in the broad sense.

The key areas of interest run across the intersection of common research areas of the MATHS and COMSC groups. We are interested in combinatorial, geometric, computational, algorithmic, and optimisation aspects of the mathematical research, as well as in mathematical foundations for topics in Computer Science.

## Aims

Our activities are focused on promotion and enhancement of collaborations between researchers from the Schools of Mathematics and Computer Science and Informatics through organisation of seminars, workshops, and discussion groups.

We aim to bring together researchers working in different areas of Discrete Mathematics and Data Science, as well as in a number of other areas within Computer Science and related Mathematics.

Our team member's research interests currently include:

• Discrete Geometry and Geometry of Numbers
• Integer Programming
• Optimisation in Graphs and Networks
• Design and Analysis of Algorithms
• Algorithm Engineering
• Data Analysis and Data Mining
• Machine Learning
• Topological Data Analysis

## Dr Iskander Aliev

Senior Lecturer

Email:
alievi@cardiff.ac.uk
Telephone:
+44 (0)29 2087 5547

## Dr Andrei Gagarin

Lecturer in Mathematics

Email:
gagarina@cardiff.ac.uk
Telephone:
+44 (0)29 2068 8850

Lecturer

Email:
corcoranp@cardiff.ac.uk
Telephone:
+44 (0)29 2087 6996

## Dr Angela Mihai

Senior Lecturer in Applied Mathematics

Email:
mihaila@cardiff.ac.uk
Telephone:
+44 (0)29 2087 5570

## Dr Thomas Woolley

Lecturer in Applied Mathematics

Email:
woolleyt1@cardiff.ac.uk
Telephone:
02920 870618

## Professor Anatoly Zhigljavsky

Chair in Statistics

Email:
zhigljavskyaa@cardiff.ac.uk
Telephone:
+44 (0)29 2087 5076

## Dr Andrey Pepelyshev

Lecturer

Email:
pepelyshevan@cardiff.ac.uk
Telephone:
+44 (0)29 2087 5530

## Seminars

All seminars are held at 15:10-16:00 in Room M/1.02, Senghennydd Road, Cardiff unless stated otherwise.

Programme Coordinators:

Dr Iskander Aliev (MATHS), Dr P. Corcoran (COMPS) and Dr A. Gagarin (MATHS/COMPS)

DateSpeakerSeminar

12 December 2017

Prof. Anthony C. Atkinson (LSE)

Robust Constrained Clustering

In cluster analysis the data are divided into k groups of similar individuals, with k a parameter to be estimated. In model-based clustering (we assume clusters of normally distributed observations) the likelihood of the observations is maximised for a range of values of k. A penalty for complexity, typically the Bayesian Information Criterion (BIC), is then used to choose the number of clusters.

These classification/mixture likelihoods are unbounded, so it is necessary to modify the problem. We introduce a constraint that the maximal ratio between the eigenvalues of the scatter matrices be less than a constant c = 1. This avoids threadlike clusters. We introduce a new penalised likelihood criterion, generalising BIC, which takes into account the higher model complexity possible with large values of c.

We extend this analysis to outlying observations which should not be forced to belong to one of the multivariate normal clusters. This robustness is achieved by trimming a proportion a of the observations. Of course, a is unknown. We therefore monitor analyses over a range of values of a. Our aim is to select optimum values of k, c and a.

Our major example uses data on 424 cows with bovine phlegmon. In addition to presenting a final clustering, we use plots to illustrate the stability of the proposed solution.

The talk is based on a joint work with Marco Riani (University of Parma).

1 November 2017

Prof Jacek Gondzio (University of Edinburgh)

Continuation in Optimization: From Interior Point Methods to Big Data

In this talk we will discuss similarities between two homotopy-based
approaches:
1. (inexact) primal-dual interior point method for LP/QP, and
2. preconditioned Newton conjugate gradient method for big data
optimization.
Both approaches rely on clever exploitation of the curvature
of optimized functions and deliver efficient techniques
for solving optimization problems of unprecedented sizes.
We will address both theoretical and practical aspects
of these methods applied to solve various inverse problems
arising in signal processing.

Part of this work was done joitly with my former PhD student,
Kimonas Fountoulakis.

References:

J. Gondzio,
Convergence analysis of an inexact feasible
interior point method for convex quadratic programming,
SIAM Journal on Optimization 23 (2013) No 3, pp. 1510-1527.
DOI: 10.1137/120886017

J. Gondzio,
Interior point methods 25 years later,
European Journal of Operational Research 218 (2012) pp. 587--601.
DOI: 10.1016/j.ejor.2011.09.017

I. Dassios, K. Fountoulakis and J. Gondzio,
A preconditioner for a primal-dual Newton conjugate gradients method
for compressed sensing problems,
SIAM Journal on Scientific Computing 37 (2016) A2783--A2812.
DOI: 10.1137/141002062

K. Fountoulakis and J. Gondzio,
A second-order method for strongly convex $\ell_1$-regularization problems,
Mathematical Programming 156 (2016), pp. 189--219.
DOI: 10.1007/s10107-015-0875-4

24 October 2017

Bailin Deng (School of Computer Science and
Informatics at Cardiff University)

Geometry Processing for Design and Fabrication

Abstract: In the past few decades, advances in digital design tools havemade it possible to create complex 3D shapes on a computer. However, physical realisation of these shapes remains a challenging task. Recently, the emergence of affordable fabrication tools such as 3Dprinters and laser cutters allows us to turn a digital design into aphysical object. But effective use of these tools requires the designshape to satisfy certain requirements related to the fabricationtechnologies, which are not considered by traditional 3D design tools.We argue that these fabrication requirements can be incorporated intothe design process as geometric constraints, such that the resultingdesigns can be realised using designated technologies and materials. Wewill present a few fabrication-aware design tools for differentapplications, and show how geometry knowledge can help to solvechallenging design problems.

22 June 2017

From which world is your graph?

Discovering statistical structure from links is a fundamental problem in the analysis of social networks. Choosing a misspecified model, or equivalently, an incorrect inference algorithm will result in an invalid analysis or even falsely uncover patterns that are in fact artifacts of the model. This work focuses on unifying two of the most widely used link-formation models: the stochastic blockmodel (SBM) and the small world (or latent space) model (SWM). Integrating techniques from kernel learning, spectral graph theory, and nonlinear dimensionality reduction, we develop the first statistically sound polynomial-time algorithm to discover latent patterns in sparse graphs for both models. When the network comes from an SBM, the algorithm outputs a block structure. When it is from an SWM, the algorithm outputs estimates of each node's latent position. We report experiments performed on a twitter dataset collected between October and November 2016.

Based on joint work with Cheng Li, Zhenming Liu and Felix Wong.

1 June 2017

Dr Peter Richtarik (University of Edinburgh)

Stochastic reformulations of linear systems and efficient
randomized algorithms

Please note different time and venue: 15:00 in WX3.07 (Queens Buildings).

We propose a new paradigm for solving linear systems with a
very large number of equations. In our paradigm, the system is first
reformulated into a stochastic problem, and then solved with a suitable (typically randomized) algorithm. Our stochastic reformulation is flexible as it depends on a user-defined parameter in the form of a distribution defining an ensemble of random matrices. The choice of the distribution directly influences the “condition number” of the reformulation, which leads to the novel concept of “randomized preconditioning”. We give necessary and sufficient conditions for the reformulation to be exact, i.e., for the solution set of the stochastic problem to be identical to the solution set of the linear system. We also show that the reformulation can be equivalently seen as a stochastic optimization problem, stochastically preconditioned linear system, stochastic fixed-point problem and as a probabilistic intersection problem. For instance, the condition number of the reformulation is equal to the condition number of the stochastically preconditioned linear system, and to the condition number of associated with the Hessian of the objective function appearing the stochastic optimization reformulation.

Further, we propose and analyze basic, parallel and accelerated
stochastic algorithms for solving the reformulated problem, with linear
convergence rates. The methods have natural and sometimes surprising interpretations from the viewpoint of each of the four reformulations.For instance, the methods can be interpreted as basic, parallel and accelerated variants of stochastic gradient descent, stochastic Newton descent, stochastic projection method and stochastic fixed-point method. The complexity of the basic variants scales linearly with the condition number of the reformulation, while the accelerated variants scale with the square root of the condition number. Moreover, all our methods lend themselves to a natural dual interpretation as “stochastic subspace ascent” methods, a  novel class of optimization algorithms not analyzed before. Stochastic dual coordinate ascent and stochastic dual Newton ascent arise in special cases. We prove global linear convergence of all our algorithms. Further, we highlight a close connection to recent algorithmic developments in machine learning through casting the problem
as an instance of the Empirical Risk Minimization problem in a new
regime not studied before.

The above development can be extended to matrix inversion. In particular, we develop and analyze a broad family of stochastic/randomized algorithms for inverting a matrix, with specialized variants maintaining symmetry and/or positive definiteness of the iterates. All methods in the family converge globally and linearly, with explicit rates. In special cases, we obtain stochastic block variants of several quasi-Newton updates, including bad Broyden (BB), good Broyden (GB), Powell-symmetric-Broyden (PSB), Davidon-Fletcher-Powell (DFP) and Broyden-Fletcher-Goldfarb-Shanno (BFGS). Ours are the first stochastic versions of these updates shown to converge to an inverse of a fixed matrix. Through a dual viewpoint we uncover a fundamental link between quasi-Newton updates and approximate inverse preconditioning. Further, we develop an adaptive variant of randomized block BFGS, where we modify the distribution underlying the stochasticity of the method throughout the iterative process to achieve faster convergence. Further, for rectangular and non-invertible matrices, variants of our methods can be shown to converge to the Moore-Penrose pseudoinverse.

28 April 2017

Richard Booth (School of Computer Science and Informatics, Cardiff University)

Quantifying distance between argumentation graph-labellings

An argumentation framework can be seen as a graph that expresses, in an abstract way, the conflicting information of an underlying logical knowledge base. The nodes of the graph correspond to arguments, and the edges correspond to attacks between arguments. This conflicting information often allows for the presence of more than one possible reasonable position which one can take, with each position corresponding to a particular labelling of the graph that labels arguments “accepted”, “rejected” or “undecided”. A relevant question, therefore, is how much these positions differ from each other. In this talk, we will examine the issue of how to define meaningful measures of distance between the (complete) labellings of a given argumentation framework. We provide concrete distance measures based on argument-wise label difference, as well as based on the notion of *issues*, and examine their properties.

24 March 2017

Dr Frank Langbein (School of Computer Science and Informatics, Cardiff University)

Robust Control of Quantum Spin Networks

Networks of spin-1/2 particles form basic components of quantum devices for networking, simulation and computing. Information stored in spin states can propagate through such networks without any charge transport in a wave-like manner, dispersing and refocusing over the network. We specifically study single excitation dynamics in spin rings and chains. The resulting traffic shows surprising non-classical behaviour, such as the presence of an anti-core, a node with minimal traffic instead of the congestion core in traditional networks. Optimal control of quantum systems can be utilised to direct this traffic. In particular shaping the energy landscape via static bias fields, instead of the usual dynamic modulation or switching of control fields, yields highly efficient, simple control schemes.

Due to the limited degrees of freedom, finding such controls via optimisation is more difficult than for dynamic control. Nevertheless, high-fidelity results can be achieved. Some of the resulting controls show the astounding property of being optimally robust at optimal performance, as demonstrated by log-sensitivity results and the rank correlation between fidelity and the mu of the structured singular value analysis.

For practical implementation of such control schemes, network topologies and couplings must be identified from measurements of specific devices to build a model suitable to find the necessary controls. For this we present an approach for discriminating between different network structures and learning model parameters. Overall this holds the promise to achieve robust, scalable quantum devices, such as routers and memories, complex spin dynamics simulators, towards computing devices.

17 March 2017

Andrei Gagarin (Cardiff)

The bondage number of graphs on topological surfaces and Teschner’s conjecture

The domination number of a graph is the smallest number of its vertices which have all the other vertices in their neighborhoods. The bondage number of a graph is the smallest number of its edges whose removal results in a graph having a larger domination number. In a sense, the bondage number measures integrity and reliability of the smallest dominating sets with respect to edge removals in graphs, which may correspond, e.g., to link failures in communication networks. The decision problem for the bondage number is known to be NP-hard.

We provide constant upper bounds for the bondage number with respect to embeddability of graphs on topological surfaces, and improve upper bounds for the bondage number in terms of the maximum vertex degree and the orientable and non-orientable graph genera. Also, we present stronger upper bounds for graphs with no triangles and graphs with the number of vertices larger than a certain threshold in terms of graph genera. This settles Teschner’s Conjecture in the affirmative for almost all graphs. As an auxiliary result, we show tight lower bounds for the number of vertices of graphs 2-cell embeddable on topological surfaces of a given genus.
(Joint work with Vadim Zverovich, University of the West of England, Bristol, UK).

17 February 2017

Prof Owen Jones (Cardiff)

The cross-entropy method for discrete optimisation

The cross-entropy method is a versatile heuristic tool for solving difficult estimation and optimization problems, based on Kullback–Leibler (or cross-entropy) minimization. It has been successfully applied to combinatorial optimisation problems such as the travelling salesman problem, the quadratic assignment problem, and the max-cut problem. It can also be used for noisy problems such as the buffer allocation problem.

In this talk I will give an introduction to the Cross-Entropy method, and then present some theoretical results that show it can find an optimal solution with probability arbitrarily close to 1, and give conditions under which an optimal solution is generated eventually with probability 1.

25 November 2016

Dr. Shoaib Jameel (School of Computer Science and Informatics, Cardiff University)

Towards embeddings for information retrieval

We present two recent models to generate embeddings. In the first model, we concentrate on generating conceptual subspaces. Conceptual spaces are geometric representations of conceptual knowledge in which entities correspond to points, natural properties correspond to convex regions, and the dimensions of the space correspond to salient features. While conceptual spaces enable elegant models of various cognitive phenomena, the lack of automated methods for constructing such representations have so far limited their application in artificial intelligence. To address this issue, we propose a method which learns a vector-space embedding of entities from Wikipedia and constrains this embedding such that entities of the same semantic type are located in some lower-dimensional subspace. In the second part of the talk, we also propose a new word embedding model, inspired by GloVe, which is formulated as a feasible least squares optimization problem. In contrast to existing models, we explicitly represent the uncertainty about the exact definition of each word vector. To this end, we estimate the error that results from using noisy co-occurrence counts in the formulation of the model, and we model the imprecision that results from including uninformative context words. Our experimental results demonstrate that these models compare favourably with existing models.

1 November 2016

Dr. Ender Ozcan (School of Computer Science, University of Nottingham)

Improving Heuristic Optimisation via Tensor Analysis

The advanced optimisation techniques underpin many of the automated decision support systems, producing “high quality”, or the “best” solutions to the challenging real-world problems. Nevertheless, the complexity of such techniques has increased along with the complexity of problems to the point where their configuration and/or control is in itself a significant challenge. There is a growing number of studies in addressing this issue with the usage of various data science techniques. In this talk, we illustrate the potential of tensor analysis as an advanced machine learning technique for improving the performance of several heuristic optimisation methods on selected case studies, including nurse rostering.

28 October 2016

Dr. Yu-Kun Lai (Cardiff University, School of Computer Science and Informatics)

Analysis and Processing of 3D Shape Collections

With the availability of cheap 3D cameras and accessible 3D modelling packages, 3D geometric shapes have proliferated in recent years. Processing potentially large-scale shape collections effectively provides both opportunities and challenges. My talk will overview some of our recent effort in this research direction. We develop geometric modelling techniques that greatly benefit from shape collections for more meaning and intuitive interactive shape editing, shape interpolation and morphing editing. We also take on the challenge by developing techniques to allow interactive exploration of large-scale heterogeneous shape repositories using efficient shape analysis and visualisation.

30 September 2016

Prof. Anatoly Zhigljavsky (Cardiff)

Linear regression analysis with large number of predictors

14 September 2016

Prof. Igor Pak (UCLA)

How to prove Steinitz's theorem

Steinitz's theorem is a classical but very remarkable result characterising graphs of convex polytopes in R^3. In this talk, I will first survey several known proofs, and present one that is especially simple. I will then discuss the quantitative version and recent advances in this direction. Joint work with Stedman Wilson.