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Spectral graph theory

About: Spectral graph theory is a(n) research topic. Over the lifetime, 1334 publication(s) have been published within this topic receiving 77373 citation(s).

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Papers
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Open accessJournal ArticleDOI: 10.1109/34.868688
Jianbo Shi1, Jitendra Malik2Institutions (2)
Abstract: We propose a novel approach for solving the perceptual grouping problem in vision. Rather than focusing on local features and their consistencies in the image data, our approach aims at extracting the global impression of an image. We treat image segmentation as a graph partitioning problem and propose a novel global criterion, the normalized cut, for segmenting the graph. The normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups. We show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion. We applied this approach to segmenting static images, as well as motion sequences, and found the results to be very encouraging.

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  • Fig. 3. Subplot (a) plots the smallest eigenvectors of the generalized eigenvalue system (11). Subplots (b)-(i) show the eigenvectors corresponding the second smallest to the ninth smallest eigenvalues of the system. The eigenvectors are reshaped to be the size of the image.
    Fig. 3. Subplot (a) plots the smallest eigenvectors of the generalized eigenvalue system (11). Subplots (b)-(i) show the eigenvectors corresponding the second smallest to the ninth smallest eigenvalues of the system. The eigenvectors are reshaped to be the size of the image.
  • Fig. 16. A weighting function with medium rate of fall-off: w x eÿd x 0:2 , shown in subplot (a) in solid line. The dotted lines show the two alternative weighting functions used in Figs. 14 and 15. Subplot (b) shows the corresponding graph weight matrix W . The two columns (c) and (d) below show the first and second extreme eigenvectors for the Normalized cut (row 1), Average cut (row 2), and average association (row 3). All three of these algorithms perform satisfactorily in this case, with normalized cut producing a clearer solution than the other two cuts.
    Fig. 16. A weighting function with medium rate of fall-off: w x eÿd x 0:2 , shown in subplot (a) in solid line. The dotted lines show the two alternative weighting functions used in Figs. 14 and 15. Subplot (b) shows the corresponding graph weight matrix W . The two columns (c) and (d) below show the first and second extreme eigenvectors for the Normalized cut (row 1), Average cut (row 2), and average association (row 3). All three of these algorithms perform satisfactorily in this case, with normalized cut producing a clearer solution than the other two cuts.
  • Fig. 1. A case where minimum cut gives a bad partition.
    Fig. 1. A case where minimum cut gives a bad partition.
  • Fig. 4. (a) shows the original image of size 80 100. Image intensity is normalized to lie within 0 and 1. Subplots (b)-(h) show the components of the partition with Ncut value less than 0.04. Parameter setting: I 0:1, X 4:0, r 5.
    Fig. 4. (a) shows the original image of size 80 100. Image intensity is normalized to lie within 0 and 1. Subplots (b)-(h) show the components of the partition with Ncut value less than 0.04. Parameter setting: I 0:1, X 4:0, r 5.
  • Fig. 11. Subimages (a) and (b) show two frames of an image sequence. Segmentation results on this two frame image sequence are shown in subimages (c) to (g). Segments in (c) and (d) correspond to the person in the foreground and segments in (e) to (g) correspond to the background. The reason that the head of the person is segmented away from the body is that, although they have similar motion, their motion profiles are different. The head region contains 2D textures and the motion profiles are more peaked, while, in the body region, the motion profiles are more spread out. Segment (e) is broken away from (f) and (g) for the same reason.
    Fig. 11. Subimages (a) and (b) show two frames of an image sequence. Segmentation results on this two frame image sequence are shown in subimages (c) to (g). Segments in (c) and (d) correspond to the person in the foreground and segments in (e) to (g) correspond to the background. The reason that the head of the person is segmented away from the body is that, although they have similar motion, their motion profiles are different. The head region contains 2D textures and the motion profiles are more peaked, while, in the body region, the motion profiles are more spread out. Segment (e) is broken away from (f) and (g) for the same reason.
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13,025 Citations


Open accessBook
Fan Chung1Institutions (1)
03 Dec 1996-
Abstract: Eigenvalues and the Laplacian of a graph Isoperimetric problems Diameters and eigenvalues Paths, flows, and routing Eigenvalues and quasi-randomness Expanders and explicit constructions Eigenvalues of symmetrical graphs Eigenvalues of subgraphs with boundary conditions Harnack inequalities Heat kernels Sobolev inequalities Advanced techniques for random walks on graphs Bibliography Index.

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Topics: Integral graph (63%), Spectral graph theory (61%), Laplacian matrix (60%) ...read more

6,908 Citations


Open accessJournal ArticleDOI: 10.1103/PHYSREVE.74.036104
Mark Newman1Institutions (1)
11 Sep 2006-Physical Review E
Abstract: We consider the problem of detecting communities or modules in networks, groups of vertices with a higher-than-average density of edges connecting them. Previous work indicates that a robust approach to this problem is the maximization of the benefit function known as ``modularity'' over possible divisions of a network. Here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix, which plays a role in community detection similar to that played by the graph Laplacian in graph partitioning calculations. This result leads us to a number of possible algorithms for detecting community structure, as well as several other results, including a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong. The algorithms and measures proposed are illustrated with applications to a variety of real-world complex networks.

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Topics: Modularity (networks) (61%), Adjacency matrix (60%), Graph partition (60%) ...read more

4,062 Citations


Open accessProceedings Article
21 Aug 2003-
Abstract: An approach to semi-supervised learning is proposed that is based on a Gaussian random field model. Labeled and unlabeled data are represented as vertices in a weighted graph, with edge weights encoding the similarity between instances. The learning problem is then formulated in terms of a Gaussian random field on this graph, where the mean of the field is characterized in terms of harmonic functions, and is efficiently obtained using matrix methods or belief propagation. The resulting learning algorithms have intimate connections with random walks, electric networks, and spectral graph theory. We discuss methods to incorporate class priors and the predictions of classifiers obtained by supervised learning. We also propose a method of parameter learning by entropy minimization, and show the algorithm's ability to perform feature selection. Promising experimental results are presented for synthetic data, digit classification, and text classification tasks.

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Topics: Semi-supervised learning (70%), Unsupervised learning (64%), Empirical risk minimization (64%) ...read more

3,709 Citations


Open accessJournal ArticleDOI: 10.21136/CMJ.1973.101168
Topics: Algebraic graph theory (72%), Irreducible component (70%), Algebraic connectivity (67%) ...read more

3,549 Citations


Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20222
202153
202086
201981
201855
2017134

Top Attributes

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Topic's top 5 most impactful authors

Ali Kaveh

19 papers, 579 citations

Edwin R. Hancock

16 papers, 857 citations

Pierre Vandergheynst

11 papers, 6K citations

Francesco Belardo

9 papers, 100 citations

Richard C. Wilson

8 papers, 380 citations

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