Divide-and-Cluster: Spatial Decomposition Based Hierarchical Clustering

Narendra Ahuja, Akshat Sharma, Divyam Goel

Keywords: Unsupervised Learning, High-dimensional features, World Centered Clustering, Points Centered Clustering, Hierarchical Clustering, Complexity, Minimal Spanning Tree

TL;DR: This paper clusters n points located in a D-dimensional space by detecting their mutual clustering affinity within local neighborhoods, using more efficient local computations, and then hierarchically growing the local clusters outward.

Abstract: This paper is about increasing the computational efficiency of clustering algorithms. Many clustering algorithms are based on properties of relative locations of points, globally or locally, e.g., interpoint distances and nearest neighbor distances. This amounts to using a lower dimensional space than the full dimensionality $D$ of the space in which the points are embedded. We present a clustering algorithm, Divide-and-Cluster (DAC), which detects local clusters in small neighborhoods obtained by recursive tessellation of space, and then merges them hierarchically, following the Divide-and-Conquer paradigm. This significantly reduces computation time which may otherwise grow nonlinearly number $n$ of points. We define locality as hypercubical neighborhoods in a recursive hypercubical decomposition of space, represented by a tree. Clusters are detected within each hypercube, and merged with those from neighboring hypercubes while traversing up the tree. We expect DAC to perform better than many other algorithms because (a) as clusters merge into larger clusters (components), their number steadily decreases vs the number of points, and (b) we cluster only neighboring components. The ordering of component appearances also simultaneously yields a cluster hierarchy (tree). Further, our use of small neighborhoods allows piecewise uniform approximation of large, nonuniform, arbitrary shaped clusters, thus avoiding the need for global cluster models. We experimentally verify the correctness of detected clusters on a variety of datasets, posing a variety of challenges, as well as show that DAC’s runtime is significantly better than representative algorithms of other types, particularly for increasing values of $n$.

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