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Matrix Optimization in Machine Learning


   Faculty of Social Sciences

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  Dr qi Qi  No more applications being accepted  Funded PhD Project (UK Students Only)

About the Project

Optimization has become an essential part of modern machine learning and data analytics. Linear algebra and matrix analysis play an important role in developing fast, reliable and applicable machine learning algorithms in a wide range of disciplines. The project focuses on optimization methods for dimensionality reduction (DR), which is one of the most challenging and useful areas in machine learning.

Data are often embedded in very high dimensional spaces. DR methods aim to represent such high-dimensional data in a low-dimensional space and to achieve two targets: preserving local structure and revealing global separation. It is often done through distance learning among data. An important passage between distances and optimization is that distances are often represented by kernels, which are required to be positive semidefinite resulting in matrix optimization. However, matrix optimization often needs spectral techniques, which involve eigenvalue-eigenvector decompositions. This is widely regarded as a bottle neck for many matrix optimization methods. Th project aims to provide a solution to this challenge.

The key framework is the Euclidean Distance Matrix (EDM) optimization and its variants. EDM optimization has a deep root in the classical Multi-Dimensional Scaling and has recently found important applications in network localization, molecular conformation, graph drawing and data visualization. The project evolves around model building, algorithmic development, and applications. Depending on the interest of the applicants, the project can focus on one or more of the following specific topics.

  1. Subspace methods: This is to overcome the computational cost of eigenvalue-eigenvector decompositions.
  2. Distributed algorithms: As data become big, it is essential to develop distributed versions of machine learning algorithms. Stochastic gradient methods would be a starting point for this part.
  3. Supervised learning: It is one of the great advantages using matrix optimization that it can incorporate side information such as label into the model. This results in the exciting topic of supervised learning through matrix optimization.
  4. Social network visualization: An important progress is that communicability distances among the agents placed the network on a sphere. How to best represent the network on a sphere is an optimization problem on the communicability distance matrix, which is EDM.

There exists a large number of applications with any progress to the listed topics above. Depending on the interest of the applicants, one may explore its applications to graphical signal processing, graph drawing, or molecular conformation (to name a few).

It is ideal that a candidates has strong background in numerical linear algebra and good knowledge of optimization with coding skills such in Python and/or Matlab.

Main supervisor. Professor Houduo Qi, https://www.southampton.ac.uk/maths/about/staff/hdqi.page

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 be made online.

Applications should include:

Research Proposal

Curriculum Vitae

Two reference letters

Degree Transcripts to date

Apply online: https://www.southampton.ac.uk/courses/how-to-apply/postgraduate-applications.page


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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