Analytic Representation of Bayes Labeling and Bayes Clustering Operators for Random Labeled Point Processes

Abstract

Clustering algorithms typically group points based on some similarity criterion, but without reference to an underlying random process to make clustering algorithms rigorously predictive. In fact, there exists a probabilistic theory of clustering in the context of random labeled point sets in which clustering error is defined in terms of the process. In the present paper, given an underlying point process we develop a general analytic procedure for finding an optimal clustering operator, the Bayes clusterer, that corresponds to the Bayes classifier in classification theory. We provide detailed solutions under Gaussian models.Owing to computational complexity we develop approximations of the Bayes clusterer. A companion website is available at: http://ece.osu.edu/∼dalton/supplementary/2014BayesClustering.

Hierarchical Overlapping Clustering of Network Data Using Cut Metrics

“…In this setting, the general objective of clustering is to group those nodes that are more similar to each other than to the rest, according to the relationship established by the dissimilarity function [1], [2]. Clustering and its generalizations are ubiquitous tools since they are used in a wide variety of fields such as psychology [3], social network analysis [4], political science [5], neuroscience [6], among many others [7], [8].…”

“…In a strict sense, the proof of Theorem 2 does not require c X to be a cut metric. In fact, any nonnegative, symmetric function satisfying the identity property would suffice to create a relationship between nodes as in (7). Constructing such a relation is known as the Rips complex [30].…”

Optimal clustering under uncertainty

“…Our interest here is in clustering, where the underlying process is a random point set and the aim is to partition the point set into clusters corresponding to the manner in which the points have been generated by the underlying process. Having developed the theory of optimal clustering in the context of random labeled point sets where optimality is with respect to mis-clustered points [1], we now consider optimal clustering when the underlying random labeled point process belongs to an uncertainty class of random labeled point processes, so that optimization is relative to both clustering error and model uncertainty. This is analogous to finding an optimal Wiener filter when the signal process is unknown, so that the power spectra belong to an uncertainty class [2].…”

“…The exceptions, for instance, expectationmaximization based on mixture models, typically focus on parameter estimation rather than defining and minimizing a notion of operator error. Work in [9] and [1] addresses the solution to (1) in the context of clustering using a probabilistic theory of clustering for random labeled point sets and a definition of clustering error given by the expected number of "misclustered" points. This results in a Bayes clusterer, which minimizes error under the assumed probabilistic framework.…”

“…An (optimal) Bayes clusterer is analogous to an (optimal) Bayes classifier, which minimizes classification error under the assumed feature-label distribution. Here, we characterize robust clustering using the framework and definitions of error in [9] and [1], and introduce definitions of robust clustering that parallel concepts from filtering. In particular, we present minimax, MCBR and IBR clusterers, and develop effective stochastic processes for robust clustering.…”