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357
Survey of clustering data mining techniques
, 2002
"... Accrue Software, Inc. Clustering is a division of data into groups of similar objects. Representing the data by fewer clusters necessarily loses certain fine details, but achieves simplification. It models data by its clusters. Data modeling puts clustering in a historical perspective rooted in math ..."
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Cited by 408 (0 self)
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Accrue Software, Inc. Clustering is a division of data into groups of similar objects. Representing the data by fewer clusters necessarily loses certain fine details, but achieves simplification. It models data by its clusters. Data modeling puts clustering in a historical perspective rooted in mathematics, statistics, and numerical analysis. From a machine learning perspective clusters correspond to hidden patterns, the search for clusters is unsupervised learning, and the resulting system represents a data concept. From a practical perspective clustering plays an outstanding role in data mining applications such as scientific data exploration, information retrieval and text mining, spatial database applications, Web analysis, CRM, marketing, medical diagnostics, computational biology, and many others. Clustering is the subject of active research in several fields such as statistics, pattern recognition, and machine learning. This survey focuses on clustering in data mining. Data mining adds to clustering the complications of very large datasets with very many attributes of different types. This imposes unique
On Clusterings: Good, Bad and Spectral
, 2003
"... We motivate and develop a natural bicriteria measure for assessing the quality of a clustering which avoids the drawbacks of existing measures. A simple recursive heuristic is shown to have poly-logarithmic worst-case guarantees under the new measure. The main result of the paper is the analysis of ..."
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Cited by 332 (11 self)
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We motivate and develop a natural bicriteria measure for assessing the quality of a clustering which avoids the drawbacks of existing measures. A simple recursive heuristic is shown to have poly-logarithmic worst-case guarantees under the new measure. The main result of the paper is the analysis of a popular spectral algorithm. One variant of spectral clustering turns out to have effective worst-case guarantees; another finds a "good" clustering, if one exists.
Expander Flows, Geometric Embeddings and Graph Partitioning
- IN 36TH ANNUAL SYMPOSIUM ON THE THEORY OF COMPUTING
, 2004
"... We give a O( log n)-approximation algorithm for sparsest cut, balanced separator, and graph conductance problems. This improves the O(log n)-approximation of Leighton and Rao (1988). We use a well-known semidefinite relaxation with triangle inequality constraints. Central to our analysis is a ..."
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Cited by 312 (18 self)
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We give a O( log n)-approximation algorithm for sparsest cut, balanced separator, and graph conductance problems. This improves the O(log n)-approximation of Leighton and Rao (1988). We use a well-known semidefinite relaxation with triangle inequality constraints. Central to our analysis is a geometric theorem about projections of point sets in , whose proof makes essential use of a phenomenon called measure concentration.
Statistical properties of community structure in large social and information networks
"... A large body of work has been devoted to identifying community structure in networks. A community is often though of as a set of nodes that has more connections between its members than to the remainder of the network. In this paper, we characterize as a function of size the statistical and structur ..."
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Cited by 246 (14 self)
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A large body of work has been devoted to identifying community structure in networks. A community is often though of as a set of nodes that has more connections between its members than to the remainder of the network. In this paper, we characterize as a function of size the statistical and structural properties of such sets of nodes. We define the network community profile plot, which characterizes the “best ” possible community—according to the conductance measure—over a wide range of size scales, and we study over 70 large sparse real-world networks taken from a wide range of application domains. Our results suggest a significantly more refined picture of community structure in large real-world networks than has been appreciated previously. Our most striking finding is that in nearly every network dataset we examined, we observe tight but almost trivial communities at very small scales, and at larger size scales, the best possible communities gradually “blend in ” with the rest of the network and thus become less “community-like.” This behavior is not explained, even at a qualitative level, by any of the commonly-used network generation models. Moreover, this behavior is exactly the opposite of what one would expect based on experience with and intuition from expander graphs, from graphs that are well-embeddable in a low-dimensional structure, and from small social networks that have served as testbeds of community detection algorithms. We have found, however, that a generative model, in which new edges are added via an iterative “forest fire” burning process, is able to produce graphs exhibiting a network community structure similar to our observations.
Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems (Extended Abstract)
- STOC'04
, 2004
"... We present algorithms for solving symmetric, diagonally-dominant linear systems to accuracy ɛ in time linear in their number of non-zeros and log(κf (A)/ɛ), where κf (A) isthe condition number of the matrix defining the linear system. Our algorithm applies the preconditioned Chebyshev iteration with ..."
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Cited by 226 (12 self)
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We present algorithms for solving symmetric, diagonally-dominant linear systems to accuracy ɛ in time linear in their number of non-zeros and log(κf (A)/ɛ), where κf (A) isthe condition number of the matrix defining the linear system. Our algorithm applies the preconditioned Chebyshev iteration with preconditioners designed using nearly-linear time algorithms for graph sparsification and graph partitioning.
Community structure in large networks: Natural cluster sizes and the absence of large well-defined clusters
, 2008
"... A large body of work has been devoted to defining and identifying clusters or communities in social and information networks, i.e., in graphs in which the nodes represent underlying social entities and the edges represent some sort of interaction between pairs of nodes. Most such research begins wit ..."
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Cited by 208 (17 self)
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A large body of work has been devoted to defining and identifying clusters or communities in social and information networks, i.e., in graphs in which the nodes represent underlying social entities and the edges represent some sort of interaction between pairs of nodes. Most such research begins with the premise that a community or a cluster should be thought of as a set of nodes that has more and/or better connections between its members than to the remainder of the network. In this paper, we explore from a novel perspective several questions related to identifying meaningful communities in large social and information networks, and we come to several striking conclusions. Rather than defining a procedure to extract sets of nodes from a graph and then attempt to interpret these sets as a “real ” communities, we employ approximation algorithms for the graph partitioning problem to characterize as a function of size the statistical and structural properties of partitions of graphs that could plausibly be interpreted as communities. In particular, we define the network community profile plot, which characterizes the “best ” possible community—according to the conductance measure—over a wide range of size scales. We study over 100 large real-world networks, ranging from traditional and on-line social networks, to technological and information networks and
The Unique Games Conjecture, integrality gap for cut problems and embeddability of negative type metrics into `1
- In Proc. 46th IEEE Symp. on Foundations of Comp. Sci
, 2005
"... In this paper we disprove the following conjecture due to Goemans [17] and Linial [25] (also see [5, 27]): “Every negative type metric embeds into `1 with constant distortion. ” We show that for every δ> 0, and for large enough n, there is an n-point negative type metric which requires distortion ..."
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Cited by 170 (11 self)
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In this paper we disprove the following conjecture due to Goemans [17] and Linial [25] (also see [5, 27]): “Every negative type metric embeds into `1 with constant distortion. ” We show that for every δ> 0, and for large enough n, there is an n-point negative type metric which requires distortion at-least (log log n)1/6−δ to embed into `1. Surprisingly, our construction is inspired by the Unique Games Conjecture (UGC) of Khot [20], establishing a previously unsuspected connection between PCPs and the theory of metric embeddings. We first prove that the UGC implies super-constant hardness results for (non-uniform) Sparsest Cut and Minimum Uncut problems. It is already known that the UGC also implies an optimal hardness result for Maximum Cut [21]. Though these hardness results rely on the UGC, we demonstrate, nevertheless, that the corresponding PCP reductions can be used to construct “integrality gap instances ” for the respective problems. Towards this, we first construct an integrality gap instance for a natural SDP relaxation of Unique Games. Then, we “simulate ” the PCP reduction, and “translate ” the integrality gap instance of Unique Games to integrality gap instances for the respective cut problems! This enables us to prove
Relations Between Average Case Complexity and Approximation Complexity (Extended Abstract)
- In Proceedings of the 34th Annual ACM Symposium on Theory of Computing
, 2002
"... We investigate relations between average case complexity and the complexity of approximation. Our preliminary findings indicate that this is a research direction that leads to interesting insights. Under the assumption that refuting 3SAT is hard on average on a natural distribution, we derive hardne ..."
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Cited by 126 (10 self)
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We investigate relations between average case complexity and the complexity of approximation. Our preliminary findings indicate that this is a research direction that leads to interesting insights. Under the assumption that refuting 3SAT is hard on average on a natural distribution, we derive hardness of approximation results for min bisection, dense k-subgraph, max bipartite clique and the 2-catalog segmentation problem. No NP-hardness of approximation results are currently known for these problems.
Euclidean distortion and the Sparsest Cut
- In Proceedings of the 37th Annual ACM Symposium on Theory of Computing
, 2005
"... Bi-Lipschitz embeddings of finite metric spaces, a topic originally studied in geometric analysis and Banach space theory, became an integral part of theoretical computer science following work of Linial, London, and Rabinovich [29]. They presented an algorithmic version of a result of Bourgain [8] ..."
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Cited by 113 (22 self)
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Bi-Lipschitz embeddings of finite metric spaces, a topic originally studied in geometric analysis and Banach space theory, became an integral part of theoretical computer science following work of Linial, London, and Rabinovich [29]. They presented an algorithmic version of a result of Bourgain [8] which shows that every
