Multilinear objective function-based clustering

TL;DR

This paper introduces a novel clustering approach based on multilinear objective functions that generalizes traditional similarity-based methods, enabling the evaluation of fuzzy clusters and extracting the optimal number of clusters from data.

Contribution

It develops a multilinear extension framework for set functions to evaluate fuzzy clusters and demonstrates how to find optimal clusterings with an implicit determination of cluster count.

Findings

Multilinear extension enables evaluation of fuzzy and hard clusters.

The method can implicitly determine the optimal number of clusters.

The approach generalizes similarity-based clustering algorithms.

Abstract

The input of most clustering algorithms is a symmetric matrix quantifying similarity within data pairs. Such a matrix is here turned into a quadratic set function measuring cluster score or similarity within data subsets larger than pairs. In general, any set function reasonably assigning a cluster score to data subsets gives rise to an objective function-based clustering problem. When considered in pseudo-Boolean form, cluster score enables to evaluate fuzzy clusters through multilinear extension MLE, while the global score of fuzzy clusterings simply is the sum over constituents fuzzy clusters of their MLE score. This is shown to be no greater than the global score of hard clusterings or partitions of the data set, thereby expanding a known result on extremizers of pseudo-Boolean functions. Yet, a multilinear objective function allows to search for optimality in the interior of the hypercube. The proposed method only requires a fuzzy clustering as initial candidate solution, for the appropriate number of clusters is implicitly extracted from the given data set.