Unsupervised clustering under the Union of Polyhedral Cones (UOPC) model

Abstract

In this paper, we consider clustering data that is assumed to come from a union of finitely many pointed convex polyhedral cones. This model is referred to as the Union of Polyhedral Cones (UOPC) model. Similar to the Union of Subspaces (UOS) model where each data from each subspace is generated from a (unknown) basis, in the UOPC model each data from each cone is assumed to be generated from a finite number of (unknown) extreme rays. To cluster data under this model, we first build an affinity graph with the different edge weights where the edge weights are derived using a K-nearest neighbor (KNN) algorithm. Subsequently, spectral clustering is applied to obtain the clusters, and the proposed algorithm is denoted as KNN-SC. We show that on average KNN-SC outperforms Sparse Subspace Clustering by Non-negative constraints Lasso (NCL), Least squares approximation (LSA), Sparse Subspace Clustering (SSC), robust Subspace Clustering via Thresholding (TSC), and Mutual KNN based Spectral Clustering (KNNM-SC), and we provide deterministic conditions for correct clustering using KNN-SC. We show that on average KNN-SC outperforms NCL, LSA, SSC, TSC, and KNNM-SC, and we provide deterministic conditions for correct clustering using KNN-SC. For an affinity measure between the cones it is shown that as long as the cones are not very coherent and as long as the density of data within each cone exceeds a threshold, KNN-SC leads to accurate clustering. Finally, simulation results on real datasets (MNIST and CMU Motion datasets) depict that the proposed algorithm works well on real data indicating the utility of the UOPC model and the proposed algorithm.