[신간의 별자리x] 우리/미술, 그리고 ‘슬픔의 박물관’

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

In the big data era, the data are generated from different sources or observed from different views. These data are referred to as multi-view data. Unleashing the power of knowledge in multi-view data is very important in big data mining and analysis. This calls for advanced techniques that consider the diversity of different views, while fusing these data. Multi-view Clustering (MvC) has attracted increasing attention in recent years by aiming to exploit complementary and consensus information across multiple views. This paper summarizes a large number of multi-view clustering algorithms, provides a taxonomy according to the mechanisms and principles involved, and classifies these algorithms into five categories, namely, co-training style algorithms, multi-kernel learning, multiview graph clustering, multi-view subspace clustering, and multi-task multi-view clustering. Therein, multi-view graph clustering is further categorized as graph-based, network-based, and spectral-based methods. Multi-view subspace clustering is further divided into subspace learning-based, and non-negative matrix factorization-based methods. This paper does not only introduce the mechanisms for each category of methods, but also gives a few examples for how these techniques are used. In addition, it lists some publically available multi-view datasets. Overall, this paper serves as an introductory text and survey for multi-view clustering.

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Multi-view Clustering: A Survey

Given a graph G and a vertex q ∈ G, the community search query returns a subgraph of G that contains vertices related to q. Communities, which are prevalent in attributed graphs such as social networks and knowledge bases, can be used in emerging applications such as product advertisement and setting up of social events. In this paper, we investigate the attributed community query (or ACQ), which returns an attributed community (AC) for an attributed graph. The AC is a subgraph of G, which satisfies both structure cohesiveness (i.e., its vertices are tightly connected) and keyword cohesiveness (i.e., its vertices share common keywords). The AC enables a better understanding of how and why a community is formed (e.g., members of an AC have a common interest in music, because they all have the same keyword "music"). An AC can be "personalized"; for example, an ACQ user may specify that an AC returned should be related to some specific keywords like "research" and "sports".To enable efficient AC search, we develop the CL-tree index structure and three algorithms based on it. We evaluate our solutions on four large graphs, namely Flickr, DBLP, Tencent, and DBpedia. Our results show that ACs are more effective and efficient than existing community retrieval approaches. Moreover, an AC contains more precise and personalized information than that of existing community search and detection methods.

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Effective community search for large attributed graphs

With the advance of technology, data are often with multiple modalities or coming from multiple sources. Multi-view clustering provides a natural way for generating clusters from such data. Although multi-view clustering has been successfully applied in many applications, most of the previous methods assumed the completeness of each view (i.e., each instance appears in all views). However, in real-world applications, it is often the case that a number of views are available for learning but none of them is complete. The incompleteness of all the views and the number of available views make it difficult to integrate all the incomplete views and get a better clustering solution. In this paper, we propose MIC (Multi-Incomplete-view Clustering), an algorithm based on weighted nonnegative matrix factorization with L2,1 regularization. The proposed MIC works by learning the latent feature matrices for all the views and generating a consensus matrix so that the difference between each view and the consensus is minimized. MIC has several advantages comparing with other existing methods. First, MIC incorporates weighted nonnegative matrix factorization, which handles the missing instances in each incomplete view. Second, MIC uses a co-regularized approach, which pushes the learned latent feature matrices of all the views towards a common consensus. By regularizing the disagreement between the latent feature matrices and the consensus, MIC can be easily extended to more than two incomplete views. Third, MIC incorporates L2,1 regularization into the weighted nonnegative matrix factorization, which makes it robust to noises and outliers. Forth, an iterative optimization framework is used in MIC, which is scalable and proved to converge. Experiments on real datasets demonstrate the advantages of MIC.

https://link.springer.com/content/pdf/10.1007%2F978-3-319-23528-8_20.pdf

Multiple incomplete views clustering via weighted nonnegative matrix factorization with L 2,1 regularization

Given a large network, local community detection aims at finding the community that contains a set of query nodes and also maximizes (minimizes) a goodness metric. This problem has recently drawn intense research interest. Various goodness metrics have been proposed. However, most existing metrics tend to include irrelevant subgraphs in the detected local community. We refer to such irrelevant subgraphs as free riders. We systematically study the existing goodness metrics and provide theoretical explanations on why they may cause the free rider effect. We further develop a query biased node weighting scheme to reduce the free rider effect. In particular, each node is weighted by its proximity to the query node. We define a query biased density metric to integrate the edge and node weights. The query biased densest subgraph, which has the largest query biased density, will shift to the neighborhood of the query nodes after node weighting. We then formulate the query biased densest connected subgraph (QDC) problem, study its complexity, and provide efficient algorithms to solve it. We perform extensive experiments on a variety of real and synthetic networks to evaluate the effectiveness and efficiency of the proposed methods.

/pdf/robust-local-community-detection-on-free-rider-effect-and-182yujdmb7.pdf

Robust local community detection: on free rider effect and its elimination

Multi-view graph clustering aims to enhance clustering performance by integrating heterogeneous information collected in different domains. Each domain provides a different view of the data instances. Leveraging cross-domain information has been demonstrated an effective way to achieve better clustering results. Despite the previous success, existing multi-view graph clustering methods usually assume that different views are available for the same set of instances. Thus instances in different domains can be treated as having strict one-to-one relationship. In many real-life applications, however, data instances in one domain may correspond to multiple instances in another domain. Moreover, relationships between instances in different domains may be associated with weights based on prior (partial) knowledge. In this paper, we propose a flexible and robust framework, CGC (Co-regularized Graph Clustering), based on non-negative matrix factorization (NMF), to tackle these challenges. CGC has several advantages over the existing methods. First, it supports many-to-many cross-domain instance relationship. Second, it incorporates weight on cross-domain relationship. Third, it allows partial cross-domain mapping so that graphs in different domains may have different sizes. Finally, it provides users with the extent to which the cross-domain instance relationship violates the in-domain clustering structure, and thus enables users to re-evaluate the consistency of the relationship. Extensive experimental results on UCI benchmark data sets, newsgroup data sets and biological interaction networks demonstrate the effectiveness of our approach.

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Flexible and robust co-regularized multi-domain graph clustering

Given a large graph and a query node, finding its k-nearest-neighbor (kNN) is a fundamental problem. Various random walk based measures have been developed to measure the proximity (similarity) between nodes. Existing algorithms for the random walk based top-k proximity search can be categorized as global and local methods based on their search strategies. Global methods usually require an expensive precomputing step. By only searching the nodes near the query node, local methods have the potential to support more efficient query. However, most existing local search methods cannot guarantee the exactness of the solution. Moreover, they are usually designed for specific proximity measures. Can we devise an efficient local search method that applies to different measures and also guarantees result exactness? In this paper, we present FLoS (Fast Local Search), a unified local search method for efficient and exact top-k proximity query in large graphs. FLoS is based on the no local optimum property of proximity measures. We show that many measures have no local optimum. Utilizing this property, we introduce several simple operations on transition probabilities, which allow developing lower and upper bounds on the proximity. The bounds monotonically converge to the exact proximity when more nodes are visited. We further show that FLoS can also be applied to measures having local optimum by utilizing relationship among different measures. We perform comprehensive experiments to evaluate the efficiency and applicability of the proposed method.

Fast and unified local search for random walk based k-nearest-neighbor query in large graphs

A printed document can provide intrinsic features of the printer so as to distinguish which printer it comes from. But how to extract the intrinsic features is critical in printer forensics. In this paper, the page document's geometric distortion is extracted as the intrinsic features, and a printer forensics method based on the distortion is proposed. Firstly projective transformation model is used to model the geometric distortion. After the feature point set of the model is extracted, the model's parameters considered as the geometric distortion features can be estimated, and then the model's error pattern can be obtained. During the process, the least squares method is used to estimate the model's parameters, and SVM technique is used for classification. The effectiveness of the model's parameters in the printer forensics is demonstrated by experimental results.

Printer forensics based on page document's geometric distortion

Finding dense subgraphs is an important problem that has recently attracted a lot of interests. Most of the existing work focuses on a single graph (or network1). In many real-life applications, however, there exist dual networks, in which one network represents the physical world and another network represents the conceptual world. In this paper, we investigate the problem of finding the densest connected subgraph (DCS) which has the largest density in the conceptual network and is also connected in the physical network. Such pattern cannot be identified using the existing algorithms for a single network. We show that even though finding the densest subgraph in a single network is polynomial time solvable, the DCS problem is NP-hard. We develop a two-step approach to solve the DCS problem. In the first step, we effectively prune the dual networks while guarantee that the optimal solution is contained in the remaining networks. For the second step, we develop two efficient greedy methods based on different search strategies to find the DCS. Different variations of the DCS problem are also studied. We perform extensive experiments on a variety of real and synthetic dual networks to evaluate the effectiveness and efficiency of the developed methods.

Yubao Wu

Papers

Robust local community detection: on free rider effect and its elimination

Flexible and robust co-regularized multi-domain graph clustering

Fast and unified local search for random walk based k-nearest-neighbor query in large graphs

Printer forensics based on page document's geometric distortion

Finding dense and connected subgraphs in dual networks