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## Multi-level Approximate Spectral Clustering

Citations: | 3 - 0 self |

### Citations

3787 | Normalized cuts and image segmentation.
- Shi, Malik
- 2000
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Citation Context ...TRODUCTION Cluster analysis has been extensively applied to information discovery, text mining, Web analytics, and bioinformatics [1], [2]. Spectral clustering is a well-known graphtheoretic approach =-=[3]-=-, which has gained popularity for its flexibility and ability to capture non-convex geometries of datasets. Despite these desirable properties, spectral clustering is prohibitively complex for analyzi... |

603 | Cluster ensembles – a knowledge reuse framework for combining multiple partitions.
- Strehl, Ghosh
- 2002
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Citation Context ...röm and KASP. To execute k-means, we use random initialization and 500 iterations. All our comparisons are conducted by using the following evaluation metrics: 1) Normalized mutual information (NMI) =-=[18]-=-: The NMI value is computed from the confusion matrix based on the true and predicted cluster labels, ranging in [0, 1]. A high NMI value indicates that the predicted clustering matches the ground tru... |

316 | Spectral grouping using the Nystrom method.
- Fowlkes, Belongie, et al.
- 2004
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Citation Context ...ncluding the preprocessing step results in heavy computational and storage overhead. The other scheme involves using low-rank approximation and allows to save memory. The methods proposed by Nyström =-=[8]-=- are based on numerical integration theory [9] and use randomly selected samples to approximate the affinity matrix, as well as the eigenvectors of the Laplacian. It has been shown that Nyström-based... |

232 | Mapreduce for machine learning on multicore - Chu, Kim, et al. - 2006 |

188 | On the Nystrom method for approximating a gram matrix for improved kernel-based learning,”
- Drineas, Mahoney
- 2005
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Citation Context ...], [12], MAS differs from them in several important ways. First, MAS provides a general framework for fast spectral clustering that works with any kernels and various lowrank approximation algorithms =-=[13]-=-. Second, instead of uniform sampling, MAS selects data samples based on a data-dependent nonuniform probability distribution, known as optimal sampling probabilities. According to [13], such sampling... |

175 | Weighted Graph Cuts without Eigenvectors: A Multilevel Approach,
- Dhillon, Guan, et al.
- 1957
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Citation Context ...processing to reduce the size of a dataset. For example, the fast approximate spectral clustering by Yan et al. [6] applies k-means algorithm to reduce the dimensionality of a dataset. Dhillon et al. =-=[7]-=- proposed weighted graph cuts to address the scalability issue by generating multi-level shrunk graphs. Although it has been shown that such preprocessing minimizes the effect of dimensionality reduct... |

155 |
Numerical Treatment of Integral Equations
- Baker
- 1977
(Show Context)
Citation Context ...vy computational and storage overhead. The other scheme involves using low-rank approximation and allows to save memory. The methods proposed by Nyström [8] are based on numerical integration theory =-=[9]-=- and use randomly selected samples to approximate the affinity matrix, as well as the eigenvectors of the Laplacian. It has been shown that Nyström-based spectral clustering is empirically efficient ... |

114 | Fast Monte Carlo Algorithms for Matrices I: Approximating Matrix Multiplication.
- Drineas, Kannan, et al.
- 2006
(Show Context)
Citation Context ... is related with c by c ≥ 64kη24 , c, the number of samples, should be large. For ‖L̃ − L̃k‖2F , it equals ∑ t≥k σ 2 t (L̃), and ‖L̃L̃T − L̃subL̃ T sub‖F has the error bound as follows (THEOREM 1 in =-=[17]-=-), ‖L̃L̃T − L̃subL̃Tsub‖F ≤ ( η2 βc ) 1 2 ‖L̃‖2F , (30) Thus, with large c, ‖L̃L̃T − L̃subL̃Tsub‖F is also small. The above analysis shows that when c is large, the approximation error of the computed... |

58 | Fast approximate spectral clustering.
- Yan, Huang, et al.
- 2009
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Citation Context ...nalysis of large datasets. One scheme uses quantization and downsampling during data preprocessing to reduce the size of a dataset. For example, the fast approximate spectral clustering by Yan et al. =-=[6]-=- applies k-means algorithm to reduce the dimensionality of a dataset. Dhillon et al. [7] proposed weighted graph cuts to address the scalability issue by generating multi-level shrunk graphs. Although... |

49 | Improved nystrom low-rank approximation and error analysis. In:
- Zhang, Tsang, et al.
- 2008
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Citation Context ... segmentation [8] and clustering large datasets [6]. However, with random sampling, one cannot obtain the explicit connection between the reduction of data size and approximation accuracy. Kai et al. =-=[10]-=- addressed this issue by proposing an improved Nyström spectral clustering, in which the quantization error of sampling is minimized by choosing k-means cluster centers as representative points. More... |

38 | On evolutionary spectral clustering,‖
- Chi, Song, et al.
- 2009
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Citation Context ...al spectral clustering algorithm unsuitable for running on a single workstation. Distributed and cloud computing environments can be an alternative, however deployment of parallel spectral clustering =-=[5]-=- is also prohibitive due to nonlinear scalability with O(n2) space complexity. Many schemes have been previously proposed to reduce computational and space complexity of spectral clustering to enable ... |

30 | Colibri: fast mining of large static and dynamic graphs. In:
- Tong, Papadimitriou, et al.
- 2008
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Citation Context ... with the linear kernel as, S̃ = RT0 M T 0 T T 0 T0M0R0 + E. (12) To efficiently select the representative subspace and generate the compact approximation of data points, we employ the Colibri method =-=[16]-=-, which samples data points with a sampling distribution optimal in the feature space. We state, Proposition 2. Define pi = ‖A(:, i)‖2∑n j=1 ‖A(:, j)‖2 , i ∈ [1, n], (13) where A(:, i) is the ith colu... |

16 | An experimental evaluation of a montecarlo algorithm for singular value decomposition
- Drineas, Drinea, et al.
- 2003
(Show Context)
Citation Context ...ous sampling strategies can be utilized, subject to∑n i=1 pi = 1 and pi ≥ 0. The simplest one is to use uniform sampling, according to which the samples are selected with equal probability (pi = 1n ) =-=[15]-=-. In addition, [13] proposes data-dependent nonuniform probability distribution, i.e., pi = ‖S(:, i)‖2∑n j=1 ‖S(:, j)‖2 , (4) as the optimal sampling probabilities with respect to approximating S. Tha... |

4 | A randomized singular value decomposition algorithm for image processing applications
- Drinea, Drineas, et al.
(Show Context)
Citation Context ...ire process generally requires O(n2) space and O(n3) time. In many cases, only a few top eigenvectors are needed. Thus, approaches such as subspace iteration, Krylov based methods, and randomized SVD =-=[4]-=- have been proposed to gain a lower time complexity. In the case that n is 100, 000, there will be ten billion elements in the similarity matrix, requiring about 37GB memory for storage. Computing of ... |

4 | Fast monte-carlo algorithms for matrices ii: Computing low-rank approximations to a matrix - Drineas, Kannan, et al. - 2006 |

2 | A dirichlet multinomial mixture modelbased approach for short text clustering
- Yin, Wang
(Show Context)
Citation Context ...to several wellknown approximate spectral clustering algorithms. I. INTRODUCTION Cluster analysis has been extensively applied to information discovery, text mining, Web analytics, and bioinformatics =-=[1]-=-, [2]. Spectral clustering is a well-known graphtheoretic approach [3], which has gained popularity for its flexibility and ability to capture non-convex geometries of datasets. Despite these desirabl... |

2 | Workload-aware indexing for keyword search in social networks
- Bjørklund, Götz, et al.
(Show Context)
Citation Context ...veral wellknown approximate spectral clustering algorithms. I. INTRODUCTION Cluster analysis has been extensively applied to information discovery, text mining, Web analytics, and bioinformatics [1], =-=[2]-=-. Spectral clustering is a well-known graphtheoretic approach [3], which has gained popularity for its flexibility and ability to capture non-convex geometries of datasets. Despite these desirable pro... |

2 |
Multi-level Low-rank Approximationbased Spectral Clustering for image segmentation
- LJ, Dong
(Show Context)
Citation Context ...imation to the graph’s Laplacian matrix, and finally to the eigenvectors. Although several methods for approximate spectral clustering with low-rank matrix approximation have been previously proposed =-=[11]-=-, [12], MAS differs from them in several important ways. First, MAS provides a general framework for fast spectral clustering that works with any kernels and various lowrank approximation algorithms [... |