Rand index

Currently, most research on nonnegative matrix factorization (NMF) focus on 2-factor X = FG^T factorization. We provide a systematic analysis of 3-factor X = FSG^T NMF. While unconstrained 3-factor NMF is equivalent to unconstrained 2-factor NMF, constrained 3-factor NMF brings new features to constrained 2-factor NMF. We study the orthogonality constraint because it leads to rigorous clustering interpretation. We provide new rules for updating F,S,G and prove the convergence of these algorithms. Experiments on 5 datasets and a real world case study are performed to show the capability of bi-orthogonal 3-factor NMF on simultaneously clustering rows and columns of the input data matrix. We provide a new approach of evaluating the quality of clustering on words using class aggregate distribution and multi-peak distribution. We also provide an overview of various NMF extensions and examine their relationships.

We present V-measure, an external entropy-based cluster evaluation measure. V-measure provides an elegant solution to many problems that affect previously defined cluster evaluation measures including 1) dependence on clustering algorithm or data set, 2) the “problem of matching”, where the clustering of only a portion of data points are evaluated and 3) accurate evaluation and combination of two desirable aspects of clustering, homogeneity and completeness. We compare V-measure to a number of popular cluster evaluation measures and demonstrate that it satisfies several desirable properties of clustering solutions, using simulated clustering results. Finally, we use V-measure to evaluate two clustering tasks: document clustering and pitch accent type clustering.

Information theoretic measures form a fundamental class of measures for comparing clusterings, and have recently received increasing interest. Nevertheless, a number of questions concerning their properties and inter-relationships remain unresolved. In this paper, we perform an organized study of information theoretic measures for clustering comparison, including several existing popular measures in the literature, as well as some newly proposed ones. We discuss and prove their important properties, such as the metric property and the normalization property. We then highlight to the clustering community the importance of correcting information theoretic measures for chance, especially when the data size is small compared to the number of clusters present therein. Of the available information theoretic based measures, we advocate the normalized information distance (NID) as a general measure of choice, for it possesses concurrently several important properties, such as being both a metric and a normalized measure, admitting an exact analytical adjusted-for-chance form, and using the nominal [0, 1] range better than other normalized variants.