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

TL;DR

This paper introduces the UOPC model for clustering data from convex polyhedral cones, proposing algorithms like KNN, NCL, and LSA, with KNN showing superior performance under certain conditions, validated on real datasets.

Contribution

The paper formulates the UOPC model for clustering and analyzes the effectiveness of various algorithms, highlighting KNN's advantages and providing deterministic conditions for correct clustering.

Findings

KNN outperforms NCL and LSA in clustering accuracy.

Deterministic conditions for successful clustering with KNN are established.

Proposed algorithms effectively cluster real datasets like MNIST and YaleFace.

Abstract

In this paper, we consider clustering data that is assumed to come from one 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) \emph{extreme rays}.To cluster data under this model, we consider several algorithms - (a) Sparse Subspace Clustering by Non-negative constraints Lasso (NCL), (b) Least squares approximation (LSA), and (c) K-nearest neighbor (KNN) algorithm to arrive at affinity between data points. Spectral Clustering (SC) is then applied on the resulting affinity matrix to cluster data into different polyhedral cones. We show that on an average KNN outperforms both NCL and LSA and for this algorithm we provide the deterministic conditions for correct clustering. 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 leads to accurate clustering. Finally, simulation results on real datasets (MNIST and YaleFace datasets) depict that the proposed algorithm works well on real data indicating the utility of the UOPC model and the proposed algorithm.