Projection methods for clustering and semi-supervised classification

This thesis focuses on data projection methods for the purposes of clustering and semi-supervised classification, with a primary focus on clustering. A number of contributions are presented which address this problem in a principled manner; using projection pursuit formulations to identify subspaces which contain useful information for the clustering task. Projection methods are extremely useful in high dimensional applications, and situations in which the data contain irrelevant dimensions which can be counterinformative for the clustering task. The final contribution addresses high dimensionality in the context of a data stream. Data streams and high dimensionality have been identified as two of the key challenges in data clustering. The first piece of work is motivated by identifying the minimum density hyperplane separator in the finite sample setting. This objective is directly related to the problem of discovering clusters defined as connected regions of high data density, which is a widely adopted definition in non-parametric statistics and machine learning. A thorough investigation into the theoretical aspects of this method, as well as the practical task of solving the associated optimisation problem efficiently is presented. The proposed methodology is applied to both clustering and semi-supervised classification problems, and is shown to reliably find low density hyperplane separators in both contexts. The second and third contributions focus on a different approach to clustering based on graph cuts. The minimum normalised graph cut objective has gained considerable attention as relaxations of the objective have been developed, which make them solvable for reasonably well sized problems. This has been adopted by the highly popular spectral clustering methods. The second piece of work focuses on identifying the optimal subspace in which to perform spectral clustering, by minimising the second eigenvalue of the graph Laplacian for a graph defined over the data within that subspace. A rigorous treatment of this objective is presented, and an algorithm is proposed for its optimisation. An approximation method is proposed which allows this method to be applied to much larger problems than would otherwise be possible. An extension of this work deals with the spectral projection pursuit method for semi-supervised classification. iii The third body of work looks at minimising the normalised graph cut using hyperplane separators. This formulation allows for the exact normalised cut to be computed, rather than the spectral relaxation. It also allows for a computationally efficient method for optimisation. The asymptotic properties of the normalised cut based on a hyperplane separator are investigated, and shown to have similarities with the clustering objective based on low density separation. In fact, both the methods in the second and third works are shown to be connected with the first, in that all three have the same solution asymptotically, as their relative scaling parameters are reduced to zero. The final body of work addresses both problems of high dimensionality and incremental clustering in a data stream context. A principled statistical framework is adopted, in which clustering by low density separation again becomes the focal objective. A divisive hierarchical clustering model is proposed, using a collection of low density hyperplanes. The adopted framework provides well founded methodology for determining the number of clusters automatically, and also identifying changes in the data stream which are relevant to the clustering objective. It is apparent that no existing methods can make both of these claims.