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Miscellaneous Aspects of Machine Learning/Unsupervised and Semi-supervised Learning

Annie Gray, Alexander Modell, Patrick Rubin-Delanchy, Nick Whiteley

In this paper we offer a new perspective on the well established agglomerative clustering algorithm, focusing on recovery of hierarchical structure. We recommend a simple variant of the standard algorithm, in which clusters are merged by maximum average dot product and not, for example, by minimum distance or within-cluster variance. We demonstrate that the tree output by this algorithm provides a bona fide estimate of generative hierarchical structure in data, under a generic probabilistic graphical model. The key technical innovations are to understand how hierarchical information in this model translates into tree geometry which can be recovered from data, and to characterise the benefits of simultaneously growing sample size and data dimension. We demonstrate superior tree recovery performance with real data over existing approaches such as UPGMA, Ward's method, and HDBSCAN.

Miscellaneous Aspects of Machine Learning/Unsupervised and Semi-supervised Learning

Peter Macgregor

Spectral clustering is a popular and effective algorithm designed to find $k$ clusters in a graph $G$.In the classical spectral clustering algorithm, the vertices of $G$ are embedded into $\mathbb{R}^k$ using $k$ eigenvectors of the graph Laplacian matrix.However, computing this embedding is computationally expensive and dominates the running time of the algorithm.In this paper, we present a simple spectral clustering algorithm based on a vertex embedding with $O(\log(k))$ vectors computed by the power method.The vertex embedding is computed in nearly-linear time with respect to the size of the graph, andthe algorithm provably recovers the ground truth clusters under natural assumptions on the input graph.We evaluate the new algorithm on several synthetic and real-world datasets, finding that it is significantly faster than alternative clustering algorithms,while producing results with approximately the same clustering accuracy.