Fast Monte-Carlo algorithms for finding low-rank approximations

Frieze, Alan; Kannan, Ravindran; Vempala, Santosh

doi:10.1109/sfcs.1998.743487

Cited by 327 publications

(504 citation statements)

References 9 publications

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“…Corresponding approaches yield interpretable results because they embed the data in lower dimensional spaces whose basis vectors correspond to actual data points. They are guaranteed to preserve properties such as sparseness or nonnegativity and enjoy increasing popularity in the data mining community [3,11,12,17,21,22,26,28] where they have been applied to fraud detection, fMRI segmentation, collaborative filtering, and co-clustering.…”

Section: Introductionmentioning

confidence: 99%

Deterministic CUR for Improved Large-Scale Data Analysis: An Empirical Study

Thurau¹,

Kersting

Bauckhage³

2012

Proceedings of the 2012 SIAM International Conference on Data Mining

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Low-rank approximations which are computed from selected rows and columns of a given data matrix have attracted considerable attention lately. They have been proposed as an alternative to the SVD because they naturally lead to interpretable decompositions which was shown to be successful in application such as fraud detection, fMRI segmentation, and collaborative filtering. The CUR decomposition of large matrices, for example, samples rows and columns according to a probability distribution that depends on the Euclidean norm of rows or columns or on other measures of statistical leverage. At the same time, there are various deterministic approaches that do not resort to sampling and were found to often yield factorization of superior quality with respect to reconstruction accuracy. However, these are hardly applicable to large matrices as they typically suffer from high computational costs. Consequently, many practitioners in the field of data mining have abandon deterministic approaches in favor of randomized ones when dealing with today's large-scale data sets. In this paper, we empirically disprove this prejudice. We do so by introducing a novel, linear-time, deterministic CUR approach that adopts the recently introduced Simplex Volume Maximization approach for column selection. The latter has already been proven to be successful for NMF-like decompositions of matrices of billions of entries. Our exhaustive empirical study on more than 30 synthetic and real-world data sets demonstrates that it is also beneficial for CUR-like decompositions. Compared to other deterministic CUR-like methods, it provides comparable reconstruction quality but operates much faster so that it easily scales to matrices of billions of elements. Compared to sampling-based methods, it provides competitive reconstruction quality while staying in the same run-time complexity class.

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Section: Introductionmentioning

confidence: 99%

Deterministic CUR for Improved Large-Scale Data Analysis: An Empirical Study

Thurau¹,

Kersting

Bauckhage³

2012

Proceedings of the 2012 SIAM International Conference on Data Mining

View full text Add to dashboard Cite

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“…[22], [9]) Since computing the optimal solution is expensive, often approximate solutions are used in practice (e.g [10] and [14]). Achlioptas and McSherry showed in [1] how one interested in a low rank approximation of a matrix A can get a matrixÂ that has the following properties.…”

Section: Exact Counting Methodsmentioning

confidence: 99%

Spectral Counting of Triangles in Power-Law Networks via Element-Wise Sparsification

Tsourakakis

Drineas

Michelakis

et al. 2009

2009 International Conference on Advances in Social Network Analysis and Mining

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Triangle counting is an important problem in graph mining. The clustering coefficient and the transitivity ratio, two commonly used measures effectively quantify the triangle density in order to quantify the fact that friends of friends tend to be friends themselves. Furthermore, several successful graph mining applications rely on the number of triangles.In this paper, we study the problem of counting triangles in large, power-law networks. Our algorithm, SPARSI-FYINGEIGENTRIANGLE , relies on the spectral properties of power-law networks and the Achlioptas-McSherry sparsification process. SPARSIFYINGEIGENTRIANGLE is easy to parallelize, fast and accurate.We verify the validity of our approach with several experiments in real-world graphs, where we achieve at the same time high accuracy and important speedup versus a straight-forward exact counting competitor.

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“…That is, ( . Another alternative method is the Columnsampling method [Frieze et al 2004]. It approximates the spectral decomposition of A by using the SVD decomposition on C directly: c;i = m l c and U c;i = U c .…”

Section: Computational Complexity Analysis and Speed Up Strategymentioning

confidence: 99%

Using Rich Social Media Information for Music Recommendation via Hypergraph Model

Tan

Chen

et al. 2011

Social Media Modeling and Computing

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There are various kinds of social media information, including different types of objects and relations among these objects, in music social communities such as Last.fm and Pandora. This information is valuable for music recommendation. However, there are two main challenges to exploit this rich social media information: (a) There are many different types of objects and relations in music social communities, which makes it difficult to develop a unified framework taking into account all objects and relations. (b) In these communities, some relations are much more sophisticated than pairwise relation, and thus cannot be simply modeled by a graph. We propose a novel music recommendation algorithm by using both multiple kinds of social media information and music acoustic-based content. Instead of graph, we use hypergraph to model the various objects and relations, and consider music recommendation as a ranking problem on this hypergraph. While an edge of an ordinary graph connects only two objects, a hyperedge represents a set of objects. In this way, hypergraph can be naturally used to model high-order relations.

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Fast Monte-Carlo algorithms for finding low-rank approximations

Cited by 327 publications

References 9 publications

Deterministic CUR for Improved Large-Scale Data Analysis: An Empirical Study

Deterministic CUR for Improved Large-Scale Data Analysis: An Empirical Study

Spectral Counting of Triangles in Power-Law Networks via Element-Wise Sparsification

Using Rich Social Media Information for Music Recommendation via Hypergraph Model

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