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Triangle sparsifiers
 Journal of Graph Algorithms and Applications
"... In this work, we introduce the notion of triangle sparsifiers, i.e., sparse graphs which are approximately the same to the original graph with respect to the triangle count. This results in a practical triangle counting method with strong theoretical guarantees. For instance, for unweighted graphs w ..."
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Cited by 17 (5 self)
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investigate cut and spectral sparsifiers with respect to triangle counting and show that they are not optimal. Submitted:
Spanners and Sparsifiers in Dynamic Streams
"... Linear sketching is a popular technique for computing in dynamic streams, where one needs to handle both insertions and deletions of elements. The underlying idea of taking randomized linear measurements of input data has been extremely successful in providing spaceefficient algorithms for classica ..."
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Cited by 3 (2 self)
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optimal 2k − 1 for this amount of space, we show for an appropriate k that it implies the first 2pass spectral sparsifier with n 1+o(1) bits of space. Previous constructions of spectral sparsifiers in this model with a constant number of passes would require n 1+c bits of space for a constant c> 0. We
An iterative thresholding algorithm for linear inverse problems with a sparsity constraint
, 2008
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Degree3 Treewidth Sparsifiers∗
, 2014
"... We study treewidth sparsifiers. Informally, given a graph G of treewidth k, a treewidth sparsifier H is a minor of G, whose treewidth is close to k, V (H)  is small, and the maximum vertex degree in H is bounded. Treewidth sparsifiers of degree 3 are of particular interest, as routing on nodedisj ..."
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Cited by 1 (1 self)
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We study treewidth sparsifiers. Informally, given a graph G of treewidth k, a treewidth sparsifier H is a minor of G, whose treewidth is close to k, V (H)  is small, and the maximum vertex degree in H is bounded. Treewidth sparsifiers of degree 3 are of particular interest, as routing on node
Degree3 Treewidth Sparsifiers
, 2014
"... We study treewidth sparsifiers. Informally, given a graph G of treewidth k, a treewidth sparsifier H is a minor of G, whose treewidth is close to k, V (H)  is small, and the maximum vertex degree in H is bounded. Treewidth sparsifiers of degree 3 are of particular interest, as routing on nodedisj ..."
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We study treewidth sparsifiers. Informally, given a graph G of treewidth k, a treewidth sparsifier H is a minor of G, whose treewidth is close to k, V (H)  is small, and the maximum vertex degree in H is bounded. Treewidth sparsifiers of degree 3 are of particular interest, as routing on node
SPARSIFYING PRECONDITIONER FOR SOLITON CALCULATIONS
"... ABSTRACT. We develop a robust and efficient method for soliton calculations for nonlinear Schrödinger equations. The method is based on the recently developed sparsifying preconditioner combined with Newton’s iterative method. The performance of the method is demonstrated by numerical examples of ga ..."
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ABSTRACT. We develop a robust and efficient method for soliton calculations for nonlinear Schrödinger equations. The method is based on the recently developed sparsifying preconditioner combined with Newton’s iterative method. The performance of the method is demonstrated by numerical examples
New spectral methods for ratio cut partition and clustering
 IEEE TRANS. ON COMPUTERAIDED DESIGN
, 1992
"... Partitioning of circuit netlists is important in many phases of VLSI design, ranging from layout to testing and hardware simulation. The ratio cut objective function [29] has received much attention since it naturally captures both mincut and equipartition, the two traditional goals of partitionin ..."
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Cited by 295 (17 self)
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Partitioning of circuit netlists is important in many phases of VLSI design, ranging from layout to testing and hardware simulation. The ratio cut objective function [29] has received much attention since it naturally captures both mincut and equipartition, the two traditional goals of partitioning. In this paper, we show that the second smallest eigenvalue of a matrix derived from the netlist gives a provably good approximation of the optimal ratio cut partition cost. We also demonstrate that fast Lanczostype methods for the sparse symmetric eigenvalue problem are a robust basis for computing heuristic ratio cuts based on the eigenvector of this second eigenvalue. Effective clustering methods are an immediate byproduct of the second eigenvector computation, and are very successful on the “difficult” input classes proposed in the CAD literature. Finally, we discuss the very natural intersection graph
Ranking and sparsifying a connection graph
"... Abstract. Many problems arising in dealing with highdimensional data sets involve connection graphs in which each edge is associated with both an edge weight and a ddimensional linear transformation. We consider vectorized versions of the PageRank and effective resistance which can be used as basi ..."
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Cited by 2 (0 self)
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Abstract. Many problems arising in dealing with highdimensional data sets involve connection graphs in which each edge is associated with both an edge weight and a ddimensional linear transformation. We consider vectorized versions of the PageRank and effective resistance which can be used as basic tools for organizing and analyzing complex data sets. For example, the generalized PageRank and effective resistance can be utilized to derive and modify diffusion distances for vector diffusion maps in data and image processing. Furthermore, the edge ranking of the connection graphs determined by the vectorized PageRank and effective resistance are an essential part of sparsification algorithms which simplify and preserve the global structure of connection graphs. 1
Spectral Sparsification and Spectrally Thin Trees
, 2012
"... We provide results of intensive experimental data in order to investigate the existence of spectrally thin trees and unweighted spectral sparsifiers for graphs with small expansion. In addition, we also survey and prove some partial results on the existence of spectrally thin trees on dense graphs w ..."
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We provide results of intensive experimental data in order to investigate the existence of spectrally thin trees and unweighted spectral sparsifiers for graphs with small expansion. In addition, we also survey and prove some partial results on the existence of spectrally thin trees on dense graphs
Sparsifying the Fisher Linear Discriminant by Rotation
, 2014
"... Many high dimensional classification techniques have been proposed in the literature based on sparse linear discriminant analysis (LDA). To efficiently use them, sparsity of linear classifiers is a prerequisite. However, this might not be readily available in many applications, and rotations of da ..."
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Many high dimensional classification techniques have been proposed in the literature based on sparse linear discriminant analysis (LDA). To efficiently use them, sparsity of linear classifiers is a prerequisite. However, this might not be readily available in many applications, and rotations of data are required to create the needed sparsity. In this paper, we propose a family of rotations to create the required sparsity. The basic idea is to use the principal components of the sample covariance matrix of the pooled samples and its variants to rotate the data first and to then apply an existing high dimensional classifier. This rotateandsolve procedure can be combined with any existing classifiers, and is robust against the sparsity level of the true model. We show that these rotations do create the sparsity needed for high dimensional classifications and provide theoretical understanding why such a rotation works empirically. The effectiveness of the proposed method is demonstrated by a number of simulated and real data examples, and the improvements of our method over some popular high dimensional classification rules are clearly shown.
Results 1  10
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