A Morse-theoretical clustering algorithm for annotated networks and spectral bounds for fuzzy clustering

Abstract

Given a set of objects X a clustering algorithm is a formal procedure that groups together objects which are similar and separates the ones which are not, thus mimicking the human ability to categorise and group together objects. Clustering algorithms have been growing for decades and clustering has become a standard data analytic technique for many fields. Standard clustering methods however fail to integrate object metadata, often readily available to the user, in the analysis.

We present in this thesis a novel clustering algorithm, called Morse, which integrates metadata information and Morse theory, a well-known topological theory, to reveal the "basins of attraction" induced by the metadata. The algorithm is described in its general form together with a study of its performance on the LFR benchmark model. We tested Morse in a real-world scenario and showed it helped to identify phenotypes of asthma based on blood gene expression profiles. We also looked at Morse in the axiomatic setting proposed by Kleinberg and introduce a novel axiom, Monotonic Consistency, that avoids the widely-reported problematic behaviour of Kleinberg's Consistency, and a possibility result for Monotonic Consistency given again by Morse. Furthermore, we extended Kleinberg's axiomatic setting to graph clustering and proved an impossibility result for Consistency, and a possibility result for Monotonic Consistency given again by Morse.

Lastly, we explored how a general clustering algorithm affects the structure of a graph using a graph spectral distance. In this direction, we proved two different bounds for such distance with respect a graph and its quotient graph induced by a hard partition, and generalised these results to fuzzy partitions.

More information

Identifiers

ORCID for Ruben Sanchez Garcia: orcid.org/0000-0001-6479-3028

Catalogue record

Export record