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194
Consistency of spectral clustering
, 2004
"... Consistency is a key property of statistical algorithms, when the data is drawn from some underlying probability distribution. Surprisingly, despite decades of work, little is known about consistency of most clustering algorithms. In this paper we investigate consistency of a popular family of spe ..."
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Cited by 572 (15 self)
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Consistency is a key property of statistical algorithms, when the data is drawn from some underlying probability distribution. Surprisingly, despite decades of work, little is known about consistency of most clustering algorithms. In this paper we investigate consistency of a popular family of spectral clustering algorithms, which cluster the data with the help of eigenvectors of graph Laplacian matrices. We show that one of the two of major classes of spectral clustering (normalized clustering) converges under some very general conditions, while the other (unnormalized), is only consistent under strong additional assumptions, which, as we demonstrate, are not always satisfied in real data. We conclude that our analysis provides strong evidence for the superiority of normalized spectral clustering in practical applications. We believe that methods used in our analysis will provide a basis for future exploration of Laplacianbased methods in a statistical setting.
Graph sparsification by effective resistances
 SIAM J. Comput
"... We present a nearlylinear time algorithm that produces highquality sparsifiers of weighted graphs. Given as input a weighted graph G = (V, E, w) and a parameter ǫ> 0, we produce a weighted subgraph H = (V, ˜ E, ˜w) of G such that  ˜ E  = O(n log n/ǫ 2) and for all vectors x ∈ R V (1 − ǫ) ∑ ..."
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Cited by 143 (9 self)
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We present a nearlylinear time algorithm that produces highquality sparsifiers of weighted graphs. Given as input a weighted graph G = (V, E, w) and a parameter ǫ> 0, we produce a weighted subgraph H = (V, ˜ E, ˜w) of G such that  ˜ E  = O(n log n/ǫ 2) and for all vectors x ∈ R V (1 − ǫ) ∑ (x(u) − x(v)) 2 wuv ≤ ∑ (x(u) − x(v)) 2 ˜wuv ≤ (1 + ǫ) ∑ (x(u) − x(v)) 2 wuv. (1) uv∈E uv ∈ ˜ E This improves upon the sparsifiers constructed by Spielman and Teng, which had O(n log c n) edges for some large constant c, and upon those of Benczúr and Karger, which only satisfied (1) for x ∈ {0, 1} V. We conjecture the existence of sparsifiers with O(n) edges, noting that these would generalize the notion of expander graphs, which are constantdegree sparsifiers for the complete graph. A key ingredient in our algorithm is a subroutine of independent interest: a nearlylinear time algorithm that builds a data structure from which we can query the approximate effective resistance between any two vertices in a graph in O(log n) time. uv∈E
On Social Networks and Collaborative Recommendation
"... Social network systems, like last.fm, play a significant role in Web 2.0, containing large amounts of multimediaenriched data that are enhanced both by explicit userprovided annotations and implicit aggregated feedback describing the personal preferences of each user. It is also a common tendency ..."
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Cited by 105 (1 self)
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Social network systems, like last.fm, play a significant role in Web 2.0, containing large amounts of multimediaenriched data that are enhanced both by explicit userprovided annotations and implicit aggregated feedback describing the personal preferences of each user. It is also a common tendency for these systems to encourage the creation of virtual networks among their users by allowing them to establish bonds of friendship and thus provide a novel and direct medium for the exchange of data. We investigate the role of these additional relationships in developing a track recommendation system. Taking into account both the social annotation and friendships inherent in the social graph established among users, items and tags, we created a collaborative recommendation system that effectively adapts to the personal information needs of each user. We adopt the generic framework of Random Walk with Restarts in order to provide with a more natural and efficient way to represent social networks. In this work we collected a representative enough portion of the music social network last.fm, capturing explicitly expressed bonds of friendship of the user as well as social tags. We performed a series of comparison experiments between the Random Walk with Restarts model and a userbased collaborative filtering method using the Pearson Correlation similarity. The results show that the graph model system benefits from the additional information embedded in social knowledge. In addition, the graph model outperforms the standard collaborative filtering method.
The slashdot zoo: Mining a social network with negative edges
 In WWW
, 2009
"... christian.bauckhage ..."
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A Comprehensive Survey of Neighborhoodbased Recommendation Methods.
 In Recommender Systems Handbook,
, 2011
"... Abstract Among collaborative recommendation approaches, methods based on nearestneighbors still enjoy a huge amount of popularity, due to their simplicity, their efficiency, and their ability to produce accurate and personalized recommendations. This chapter presents a comprehensive survey of neig ..."
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Cited by 69 (0 self)
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Abstract Among collaborative recommendation approaches, methods based on nearestneighbors still enjoy a huge amount of popularity, due to their simplicity, their efficiency, and their ability to produce accurate and personalized recommendations. This chapter presents a comprehensive survey of neighborhoodbased methods for the item recommendation problem. In particular, the main benefits of such methods, as well as their principal characteristics, are described. Furthermore, this document addresses the essential decisions that are required while implementing a neighborhoodbased recommender system, and gives practical information on how to make such decisions. Finally, the problems of sparsity and limited coverage, often observed in large commercial recommender systems, are discussed, and a few solutions to overcome these problems are presented.
Learning Spectral Graph Transformations for Link Prediction
"... We present a unified framework for learning link prediction and edge weight prediction functions in large networks, based on the transformation of a graph’s algebraic spectrum. Our approach generalizes several graph kernels and dimensionality reduction methods and provides a method to estimate their ..."
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Cited by 48 (5 self)
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We present a unified framework for learning link prediction and edge weight prediction functions in large networks, based on the transformation of a graph’s algebraic spectrum. Our approach generalizes several graph kernels and dimensionality reduction methods and provides a method to estimate their parameters efficiently. We show how the parameters of these prediction functions can be learned by reducing the problem to a onedimensional regression problem whose runtime only depends on the method’s reduced rank and that can be inspected visually. We derive variants that apply to undirected, weighted, unweighted, unipartite and bipartite graphs. We evaluate our method experimentally using examples from social networks, collaborative filtering, trust networks, citation networks, authorship graphs and hyperlink networks. 1.
Kron Reduction of Graphs with Applications to Electrical Networks
"... Consider a weighted undirected graph and its corresponding Laplacian matrix, possibly augmented with additional diagonal elements corresponding to selfloops. The Kron reduction of this graph is again a graph whose Laplacian matrix is obtained by the Schur complement of the original Laplacian mat ..."
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Cited by 39 (16 self)
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Consider a weighted undirected graph and its corresponding Laplacian matrix, possibly augmented with additional diagonal elements corresponding to selfloops. The Kron reduction of this graph is again a graph whose Laplacian matrix is obtained by the Schur complement of the original Laplacian matrix with respect to a specified subset of nodes. The Kron reduction process is ubiquitous in classic circuit theory and in related disciplines such as electrical impedance tomography, smart grid monitoring, transient stability assessment, and analysis of power electronics. Kron reduction is also relevant in other physical domains, in computational applications, and in the reduction of Markov chains. Related concepts have also been studied as purely theoretic problems in the literature on linear algebra. In this paper we analyze the Kron reduction process from the viewpoint of algebraic graph theory. Specifically, we provide a comprehensive and detailed graphtheoretic analysis of Kron reduction encompassing topological, algebraic, spectral, resistive, and sensitivity analyses. Throughout our theoretic elaborations we especially emphasize the practical applicability of our results to various problem setups arising in engineering, computation, and linear algebra. Our analysis of Kron reduction leads to novel insights both on the mathematical and the physical side.
Audience selection for online brand advertising: privacyfriendly social network targeting
 In KDD ’09: Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining
, 2009
"... This paper describes and evaluates privacyfriendly methods for extracting quasisocial networks from browser behavior on usergenerated content sites, for the purpose of finding good audiences for brand advertising (as opposed to click maximizing, for example). Targeting socialnetwork neighbors re ..."
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Cited by 34 (2 self)
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This paper describes and evaluates privacyfriendly methods for extracting quasisocial networks from browser behavior on usergenerated content sites, for the purpose of finding good audiences for brand advertising (as opposed to click maximizing, for example). Targeting socialnetwork neighbors resonates well with advertisers, and online browsing behavior data counterintuitively can allow the identification of good audiences anonymously. Besides being one of the first papers to our knowledge on data mining for online brand advertising, this paper makes several important contributions. We introduce a framework for evaluating brand audiences, in analogy to predictivemodeling holdout evaluation. We introduce methods for extracting quasisocial networks from data on visitations to social networking pages, without collecting any information on the identities of the browsers or the content of the socialnetwork pages. We introduce measures of brand proximity in the network, and show that audiences with high brand proximity indeed show substantially higher brand affinity. Finally, we provide evidence that the quasisocial network embeds a true social network, which along with results from social theory offers one explanation for the increases in audience brand affinity.
Inducing Crosslingual Distributed Representations of Words
, 2012
"... Distributed representations of words have proven extremely useful in numerous natural language processing tasks. Their appeal is that they can help alleviate data sparsity problems common to supervised learning. Methods for inducing these representations require only unlabeled language data, which a ..."
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Cited by 32 (1 self)
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Distributed representations of words have proven extremely useful in numerous natural language processing tasks. Their appeal is that they can help alleviate data sparsity problems common to supervised learning. Methods for inducing these representations require only unlabeled language data, which are plentiful for many natural languages. In this work, we induce distributed representations for a pair of languages jointly. We treat it as a multitask learning problem where each task corresponds to a single word, and task relatedness is derived from cooccurrence statistics in bilingual parallel data. These representations can be used for a number of crosslingual learning tasks, where a learner can be trained on annotations present in one language and applied to test data in another. We show that our representations are informative by using them for crosslingual document classification, where classifiers trained on these representations substantially outperform strong baselines (e.g. machine translation) when applied to a new language.