# Efficient methods for solving clustering-with-hyperplane problems

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Dr Stefano Coniglio  No more applications being accepted  Funded PhD Project (UK Students Only)

Clustering is one of the central problems in machine learning. In its classical paradigm, the clustering problem asks for partitioning a given dataset into k clusters in such a way that the distance between each point and the centroid of the corresponding cluster is minimized.

In many applications, the interest is in partitioning the data set in such away that the relationships of linearity and coplanarity among the point in the dataset are extracted. This entails clustering the points that belong to the same line or the same surface together and determining the analytical expression of the line or surface that best approximates them. This it the case, for instance, of digital images where one looks for a way to identify the equations of a set of lines, or of robotic vision applications where, given a sample of the 3d environment collected by a sonar, one looks for the equations of the surfaces present in the environment (such as, e.g., the floor, the ceiling, or the walls).

When tackled with mathematical-programming techniques, the problem (which is known in the literature as the hyperplane clustering problem), exhibits an interesting mathematical structure intimately related to the geometric structure of the ball determined by the set of points in Rn with unit p-norm. Such a set is, e.g., a sphere when p = 2, an octahedron when p = 1, or a cube when p = infinity (a picture can be found here: https://i.stack.imgur.com/o657Z.png). The aim of the project is investigating such a mathematical structure in order to develop efficient optimization algorithm which exploit it in a clever way. For more information on the problem, see:

https://www.sciencedirect.com/science/article/pii/S037722171200690X

The research activity lies in the intersection of mathematical optimisation and machine learning. You will build a strong background in efficient methods for solving mathematical programming problems, with focus on global-optimization algorithms for nonconvex optimization. A good understanding of calculus and linear algebra as well as some familiarity with a programming language (such as, e.g., Python) is required. Previous knowledge of branch-and-bound methods, while not required, is a plus.

### Host Institution

You will be based at the University of Southampton, a research intensive university and a founding member of the Russell Group of elite British universities. In the 2014 Research Excellence Framework, Southampton was ranked 8th for research intensity. In 2017-18, Southampton has been ranked 5th in the UK for research grant income. Besides being recognised as one of the leading research universities in the UK, Southampton has also achieved consistently high scores for its teaching and learning activities. In the Research Excellence framework, 100% of Mathematics research impact and research environment was specifically rated as of internationally excellent or world-leading quality. The broad range of Mathematical Sciences at Southampton gives Southampton a unique ability to contribute to the scientific and social challenges facing society.

Southampton has an excellent track record for optimisation. Statistics and Operational Research groups have existed within Mathematical Sciences since the 1960s. In the early 2000s, the broad multidisciplinary nature of Southampton activity in these areas was recognised through the establishment of the Centre of Operational Research, Management Sciences and Information Systems CORMSIS, which spans Mathematical Sciences and Southampton Business School. Operational Research at the University of Southampton is ranked 33th in the world, and 7th in the UK, according to the latest QS World Rankings. You will be a member of CORMSIS for the duration of your PhD studies.

### Entry Requirements

A very good undergraduate degree (at least a UK 2:1 honours degree, or its international equivalent).

### How To Apply

Apply for the research degree programme PhD Mathematical Sciences in the Faculty of Social Sciences.

### Applications should include:

Research Proposal

Curriculum Vitae

Two reference letters

Degree Transcripts to date

## Funding Notes

For UK students, Tuition Fees and a stipend of £15,285 tax-free per annum for up to 3.5 years.
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