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Stable Topological Structures in Data

  • Full or part time
  • Application Deadline
    Applications accepted all year round
  • Self-Funded PhD Students Only
    Self-Funded PhD Students Only

Project Description

The School of Mathematical Sciences of Queen Mary University of London invite applications for a PhD project commencing either in September 2019 for students seeking funding, or at any point in the academic year for self-funded students. The deadline for funded applications was 14 January 2019 (if you wish to be considered for the Alan Turing Institute studentship) or 31 January 2019 for all other funded studentships.

This project will be supervised by Dr. Primoz Skraba.

Despite the rapid development of data analysis in the last few years, a key open challenge remains in interpreting and drawing conclusions based on the results of the analysis. This is problem is exacerbated by the ``black-box" methods such as deep learning and functional methods, which while highly successful can have problematic or non-intuitive interpretations. This project will be to develop methods to find stable structures in data -- that is set-theoretic structures which are stable under some perturbation model in sensible and rigorous way. The general approach will be based on applied topology, particularly on techniques based on (persistent) (co)homology as well as strengthen connections with homotopy theory.

Stable Topological Representatives:
Algebraic invariants based on persistence, in particular persistent homology and cohomology classes, have been shown to be stable in a number of different ways. This is especially useful when these invariants are computed from finite and noisy data. Building on the stability of the algebraic invariants, the goal of this project is to understand when can these stable invariants can be stably mapped back to the underlying data through their representatives, find obstructions where this is impossible, and quantify how stable the mapping is.

Topological Constraints in Optimization: By understanding the mapping from algebraic invariants back to the underlying topological space, this project will examine applications of incorporating topological constraints into existing machine learning pipelines. The prototypical example is regularizing deep neural networks in order to promote certain global geometric features. This includes applications such as surface reconstruction with high codimension (which appear in time varying point clouds) and to deal with missing data (where topological constraints can be used to deal with the ``holes" in the data).

This project’s goal is to develop techniques to quantify instability under different models. This will look at how stability (and instability) translate across different perturbation models and how can this instability be visualized with respect to data.

The application procedure is described on the School website. For further inquiries please contact Dr. Primoz Skraba .

Funding Notes

The project can be undertaken as a self-funded project, either through your own funds or through a body external to Queen Mary University of London. Self-funded applications are accepted year-round.

The School of Mathematical Sciences is committed to the equality of opportunities and to advancing women’s careers. As holders of a Bronze Athena SWAN award we offer family friendly benefits and support part-time study. Further information is available here. We strongly encourage applications from women as they are underrepresented within the School.

We particularly welcome applicants through the China Scholarship Council Scheme.

Related Subjects

How good is research at Queen Mary University of London in Mathematical Sciences?

FTE Category A staff submitted: 34.80

Research output data provided by the Research Excellence Framework (REF)

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