Data Mining - Concepts and Techniques.

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

https://ranger.uta.edu/~chqding/papers/orthNMF.pdf

Orthogonal nonnegative matrix t-factorizations for clustering

This paper evaluates the performance of different criterion functions in the context of partitional clustering algorithms for document datasets. Our study involves a total of seven different criterion functions, three of which are introduced in this paper and four that have been proposed in the past. We present a comprehensive experimental evaluation involving 15 different datasets, as well as an analysis of the characteristics of the various criterion functions and their effect on the clusters they produce. Our experimental results show that there are a set of criterion functions that consistently outperform the rest, and that some of the newly proposed criterion functions lead to the best overall results. Our theoretical analysis shows that the relative performance of the criterion functions depends on (i) the degree to which they can correctly operate when the clusters are of different tightness, and (ii) the degree to which they can lead to reasonably balanced clusters.

/pdf/empirical-and-theoretical-comparisons-of-selected-criterion-30ahdrsx2u.pdf

Empirical and Theoretical Comparisons of Selected Criterion Functions for Document Clustering

Clustering is a division of data into groups of similar objects. Each group, called a cluster, consists of objects that are similar between themselves and dissimilar compared to objects of other groups. This paper is intended to study and compare different data clustering algorithms. The algorithms under investigation are: k-means algorithm, hierarchical clustering algorithm, self-organizing maps algorithm, and expectation maximization clustering algorithm. All these algorithms are compared according to the following factors: size of dataset, number of clusters, type of dataset and type of software used. Some conclusions that are extracted belong to the performance, quality, and accuracy of the clustering algorithms.

Comparisons Between Data Clustering Algorithms.

The nonnegative matrix factorization (NMF) has been shown recently to be useful for clustering and various extensions and variations of NMF have been proposed recently. Despite significant research progress in this area, few attempts have been made to establish the connections between various factorization methods while highlighting their differences. In this paper we aim to provide a comprehensive study on matrix factorization for clustering. In particular, we present an overview and summary on various matrix factorization algorithms and theoretically analyze the relationships among them. Experiments are also conducted to empirically evaluate and compare various factorization methods. In addition, our study also answers several previously unaddressed yet important questions for matrix factorizations including the interpretation and normalization of cluster posterior and the benefits and evaluation of simultaneous clustering. We expect our study would provide good insights on matrix factorization research for clustering.

/pdf/the-relationships-among-various-nonnegative-matrix-56sfkasinc.pdf

The Relationships Among Various Nonnegative Matrix Factorization Methods for Clustering

Bipartite Graph Partitioning and Data Clustering* Hongyuan Zha Xiaofeng He Dept. of Comp. Sci. & Eng. Penn State Univ. State College, PA 16802 {zha,xhe}@cse.psu.edu Chris Ding Horst Simon NERSC Division Berkeley National Lab. Berkeley, CA 94720 {chqding,hdsimon} Qlbl. gov Ming Gu Dept. of Math. U.C. Berkeley Berkeley, CA 94720 mgu@math.berkeley.edu ABSTRACT M a n y data types arising from data mining applications can be modeled as bipartite graphs, examples include terms and documents in a text corpus, customers and purchasing items in market basket analysis and reviewers and movies in a movie recommender system. In this paper, we propose a new data clustering method based on partitioning the underlying bipartite graph. The partition is constructed by minimizing a normalized sum of edge weights between unmatched pairs of vertices of the bipartite graph. We show that an approxi­ mate solution to the minimization problem can be obtained by computing a partial singular value decomposition ( S V D ) of the associated edge weight matrix of the bipartite graph. We point out the connection of our clustering algorithm to correspondence analysis used in multivariate analysis. We also briefly discuss the issue of assigning data objects to multiple clusters. In the experimental results, we apply our clustering algorithm to the problem of document clustering to illustrate its effectiveness and efficiency. 1. INTRODUCTION Cluster analysis is an important tool for exploratory data mining applications arising from many diverse disciplines. Informally, cluster analysis seeks to partition a given data set into compact clusters so that data objects within a clus­ ter are more similar than those in distinct clusters. The liter­ ature on cluster analysis is enormous including contributions from many research communities, (see [6, 9] for recent sur­ veys of some classical approaches.) M a n y traditional clus­ tering algorithms are based on the assumption that the given dataset consists of covariate information (or attributes) for each individual data object, and cluster analysis can be cast as a problem of grouping a set of n-dimensional vectors each representing a data object in the dataset. A familiar ex­ ample is document clustering using the vector space model [1]. Here each document is represented by an n-dimensional vector, and each coordinate of the vector corresponds to a term in a vocabulary of size n. This formulation leads to the so-called term-document matrix A = (oy) for the rep­ resentation of the collection of documents, where o y is the so-called term frequency, i.e., the number of times term i occurs in document j. In this vector space model terms and documents are treated asymmetrically with terms consid­ ered as the covariates or attributes of documents. It is also possible to treat both terms and documents as first-class citizens in a symmetric fashion, and consider a y as the fre­ quency of co-occurrence of term i and document j as is done, for example, in probabilistic latent semantic indexing [12]. In this paper, we follow this basic principle and propose a new approach to model terms and documents as vertices in a bipartite graph with edges of the graph indicating the co-occurrence of terms and documents. In addition we can optionally use edge weights to indicate the frequency of this co-occurrence. Cluster analysis for document collections in this context is based on a very intuitive notion: documents are grouped by topics, on one hand documents in a topic tend to more heavily use the same subset of terms which form a term cluster, and on the other hand a topic usually is characterized by a subset of terms and those documents heavily using those terms tend to be about that particular topic. It is this interplay of terms and documents which gives rise to what we call bi-clustering by which terms and documents are simultaneously grouped into semantically co- Categories and Subject Descriptors 11.3.3 [ I n f o r m a t i o n S e a r c h a n d R e t r i e v a l ] : Clustering; G.1.3 [ N u m e r i c a l L i n e a r A l g e b r a ] : Singular value de­ composition; G.2.2 [ G r a p h T h e o r y ] : G r a p h algorithms General Terms Algorithms, theory Keywords document clustering, bipartite graph, graph partitioning, spectral relaxation, singular value decomposition, correspon­ dence analysis *Part of this work was done while Xiaofeng He was a grad­ uate research assistant at N E R S C , Berkeley National Lab. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. CIKM '01 November 5-10, 2001, Atlanta, Georgia. U S A Copyright 2001 A C M X - X X X X X - X X - X / X X / X X ...$5.00. O u r clustering algorithm computes an approximate global optimal solution while probabilistic latent semantic indexing relies on the E M algorithm and therefore might be prone to local m i n i m a even with the help of some annealing process. x

Bipartite graph partitioning and data clustering

Many data types arising from data mining applications can be modeled as bipartite graphs, examples include terms and documents in a text corpus, customers and purchasing items in market basket analysis and reviewers and movies in a movie recommender system. In this paper, the authors propose a new data clustering method based on partitioning the underlying biopartite graph. The partition is constructed by minimizing a normalized sum of edge weights between unmatched pairs of vertices of the bipartite graph. They show that an approximate solution to the minimization problem can be obtained by computing a partial singular value decomposition (SVD) of the associated edge weight matrix of the bipartite graph. They point out the connection of their clustering algorithm to correspondence analysis used in multivariate analysis. They also briefly discuss the issue of assigning data objects to multiple clusters. In the experimental results, they apply their clustering algorithm to the problem of document clustering to illustrate its effectiveness and efficiency.

/pdf/bipartite-graph-partitioning-and-data-clustering-5fmnuxkjjn.pdf

The goal of data clustering can be formally stated as a min-max clustering principle: data points are grouped into clusters such that (a) between-cluster similarities are minimized while (b) within-cluster similarities are maximized. Existing methods typically focus on one of the above requirements. Here we propose a new method that emphasizes both requirements simultaneously. The MinMaxCut method uses pairwise similarities between all data objects. Viewing them as the adjacency matrix of a weighted graph, MinMaxCut is a graph partitioning method. An important feature is that the continuous solution of the cluster membership indicator vector is the eigenvector of the generalized graph Laplacian matrix; this simplifies the optimization of the min-max clustering objective function. We provide a comprehensive analysis of this principled clustering method; a systematic evaluation on clustering experiments on Internet newsgroups indicates that MinMaxCut outperforms other current popular partitioning/clustering methods.

A MinMaxCut Spectral Method for Data Clustering and Graph Partitioning

Many data types arising from data mining applications can be modeled as bipartite graphs, examples include terms and documents in a text corpus, customers and purchasing items in market basket analysis and reviewers and movies in a movie recommender system. In this paper, we propose a new data clustering method based on partitioning the underlying bipartite graph. The partition is constructed by minimizing a normalized sum of edge weights between unmatched pairs of vertices of the bipartite graph. We show that an approximate solution to the minimization problem can be obtained by computing a partial singular value decomposition (SVD) of the associated edge weight matrix of the bipartite graph. We point out the connection of our clustering algorithm to correspondence analysis used in multivariate analysis. We also briefly discuss the issue of assigning data objects to multiple clusters. In the experimental results, we apply our clustering algorithm to the problem of document clustering to illustrate its effectiveness and efficiency.

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Bipartite graph partitioning and data clustering

Bipartite graph partitioning and data clustering

A MinMaxCut Spectral Method for Data Clustering and Graph Partitioning

Bipartite graph partitioning and data clustering