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ACE: A Fast Multiscale Eigenvector Computation for Drawing Huge Graphs
, 2002
"... We present an extremely fast graph drawing algorithm for very large graphs, which we term ACE (for Algebraic multigrid Computation of Eigenvectors). ACE finds an optimal drawing by minimizing a quadratic energy function due to Hall, using a novel algebraic multigrid technique. The algorithm exhibits ..."
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Cited by 64 (13 self)
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We present an extremely fast graph drawing algorithm for very large graphs, which we term ACE (for Algebraic multigrid Computation of Eigenvectors). ACE finds an optimal drawing by minimizing a quadratic energy function due to Hall, using a novel algebraic multigrid technique. The algorithm exhibits an improvement of something like two orders of magnitude over the fastest algorithms we are aware of; it draws graphs of a million nodes in less than a minute. Moreover, the algorithm can deal with more general entities, such as graphs with masses and negative weights (to be defined in the text), and it appears to be applicable outside of graph drawing too.
On Spectral Graph Drawing
 Proc. 9th Inter. Computing and Combinatorics Conference (COCOON’03), LNCS 2697
, 2002
"... The spectral approach for graph visualization computes the layout of a graph using certain eigenvectors of related matrices. Some important advantages of this approach are an ability to compute optimal layouts (according to specific requirements) and a very rapid computation time. In this paper we e ..."
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Cited by 44 (10 self)
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The spectral approach for graph visualization computes the layout of a graph using certain eigenvectors of related matrices. Some important advantages of this approach are an ability to compute optimal layouts (according to specific requirements) and a very rapid computation time. In this paper we explore spectral visualization techniques and study their properties. We present a novel view of the spectral approach, which provides a direct link between eigenvectors and the aesthetic properties of the layout. In addition, we present a new formulation of the spectral drawing method with some aesthetic advantages. This formulation is accompanied by an aestheticallymotivated algorithm, which is much easier to understand and to implement than the standard numerical algorithms for computing eigenvectors.
Robust linear dimensionality reduction
 IEEE Transactions on Visualization and Computer Graphics
, 2004
"... ..."
Visualization of Labeled Data Using Linear Transformations
"... We present a novel family of datadriven linear transformations, aimed at visualizing multivariate data in a lowdimensional space in a way that optimally preserves the structure of the data. The wellstudied PCA and Fisher's LDA are shown to be special members in this family of transformations ..."
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Cited by 17 (1 self)
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We present a novel family of datadriven linear transformations, aimed at visualizing multivariate data in a lowdimensional space in a way that optimally preserves the structure of the data. The wellstudied PCA and Fisher's LDA are shown to be special members in this family of transformations, and we demonstrate how to generalize these two methods such as to enhance their performance. Furthermore, our technique is the only one, to the best of our knowledge, that reflects in the resulting embedding both the data coordinates and pairwise similarities and/or dissimilarities between the data elements. Even more so, when information on the clustering (labeling) decomposition of the data is known, this information can be integrated in the linear transformation, resulting in embeddings that clearly show the separation between the clusters, as well as their intrastructure. All this make our technique very flexible and powerful, and let us cope with kinds of data that other techniques fail to describe properly.
Drawing graphs by eigenvectors: Theory and practice
 Computers and Mathematics with Applications
, 2005
"... Abstract. The spectral approach for graph visualization computes the layout of a graph using certain eigenvectors of related matrices. Some important advantages of this approach are an ability to compute optimal layouts (according to specific requirements) and a very rapid computation time. In this ..."
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Cited by 15 (1 self)
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Abstract. The spectral approach for graph visualization computes the layout of a graph using certain eigenvectors of related matrices. Some important advantages of this approach are an ability to compute optimal layouts (according to specific requirements) and a very rapid computation time. In this paper we explore spectral visualization techniques and study their properties from different points of view. We also suggest a novel algorithm for calculating spectral layouts resulting in an extremely fast computation by optimizing the layout within a small vector space.
A TwoWay Visualization Method for Clustered Data
, 2003
"... We describe a novel approach to the visualization of hierarchical clustering that superimposes the classical dendrogram over a fully synchronized lowdimensional embedding, thereby gaining the benefits of both approaches. In a single image one can view all the clusters, examine the relations between ..."
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Cited by 10 (3 self)
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We describe a novel approach to the visualization of hierarchical clustering that superimposes the classical dendrogram over a fully synchronized lowdimensional embedding, thereby gaining the benefits of both approaches. In a single image one can view all the clusters, examine the relations between them and study many of their properties. The method is based on an algorithm for lowdimensional embedding of clustered data, with the property that separation between all clusters is guaranteed, regardless of their nature. In particular, the algorithm was designed to produce embeddings that strictly adhere to a given hierarchical clustering of the data, so that every two disjoint clusters in the hierarchy are drawn separately.
One Dimensional Layout Optimization, with Applications to Graph Drawing by Axis Separation
, 2005
"... In this paper we discuss a useful family of graph drawing algorithms, characterized by their ability to draw graphs in one dimension. We define the special requirements from such algorithms and show how several graph drawing techniques can be extended to handle this task. In particular, we suggest ..."
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Cited by 1 (1 self)
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In this paper we discuss a useful family of graph drawing algorithms, characterized by their ability to draw graphs in one dimension. We define the special requirements from such algorithms and show how several graph drawing techniques can be extended to handle this task. In particular, we suggest a novel optimization algorithm that facilitates using the Kamada and Kawai model [17] for producing onedimensional layouts. The most important application of the algorithms seems to be in achieving graph drawing by axis separation, where each axis of the drawing addresses different aspects of aesthetics.
IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS 1 Robust Linear Dimensionality Reduction
"... Abstract — We present a novel family of datadriven linear transformations, aimed at finding low dimensional embeddings of multivariate data, in a way that optimally preserves the structure of the data. The wellstudied PCA and Fisher’s LDA are shown to be special members in this family of transform ..."
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Abstract — We present a novel family of datadriven linear transformations, aimed at finding low dimensional embeddings of multivariate data, in a way that optimally preserves the structure of the data. The wellstudied PCA and Fisher’s LDA are shown to be special members in this family of transformations, and we demonstrate how to generalize these two methods such as to enhance their performance. Furthermore, our technique is the only one, to the best of our knowledge, that reflects in the resulting embedding both the data coordinates and pairwise relationships between the data elements. Even more so, when information on the clustering (labeling) decomposition of the data is known, this information can also be integrated in the linear transformation, resulting in embeddings that clearly show the separation between the clusters, as well as their internal structure. All this makes our technique very flexible and powerful, and lets us cope with kinds of data that other techniques fail to describe properly. Index Terms — Dimensionality reduction, visualization, classification, feature extraction, projection, linear transformation, principal
A TwoWay Visualization Method for Clustered Data (Extended Abstract)
"... Yehuda Koren and David Harel Dept. of Computer Science and Applied Mathematics The Weizmann Institute of Science, Rehovot, Israel {yehuda,dharel}@wisdom.weizmann.ac. il ABSTRACT We describe a novel approach to the visualization of hierarchical clustering that superimposes the classical dendrogr ..."
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Yehuda Koren and David Harel Dept. of Computer Science and Applied Mathematics The Weizmann Institute of Science, Rehovot, Israel {yehuda,dharel}@wisdom.weizmann.ac. il ABSTRACT We describe a novel approach to the visualization of hierarchical clustering that superimposes the classical dendrogram over a fully synchronized lowdimensional embedding, thereby gaining the benefits of both approaches. In a single image one can view all the clusters, examine the relations between them and study many of their properties. The method is based on an algorithm for lowdimensional embedding of clustered data, with the property that separation between all clusters is guaranteed, regardless of their nature. In particular, the algorithm was designed to produce embeddings that strictly adhere to a given hierarchical clustering of the data, so that every two disjoint clusters in the hierarchy are drawn separately.
OneDimensional Graph Drawing: Part I — Drawing Graphs by Axis Separation
"... Abstract. In this paper we discuss a useful family of graph drawing algorithms, characterized by their ability to draw graphs in one dimension. The most important application of this family seems to be achieving graph drawing by axis separation, where each axis of the drawing addresses different asp ..."
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Abstract. In this paper we discuss a useful family of graph drawing algorithms, characterized by their ability to draw graphs in one dimension. The most important application of this family seems to be achieving graph drawing by axis separation, where each axis of the drawing addresses different aspects of aesthetics. We define the special requirements from such algorithms and show how several graph drawing algorithms can be generalized to handle this task. 1