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Spectral Graph Theory Lecture 19 Fast Laplacian Solvers by Sparsification
, 2015
"... These notes are not necessarily an accurate representation of what happened in class. The notes written before class say what I think I should say. I sometimes edit the notes after class to make them way what I wish I had said. There may be small mistakes, so I recommend that you check any mathemati ..."
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mathematically precise statement before using it in your own work. These notes were last revised on November 9, 2015. 19.1 Overview We will see how sparsification allows us to solve systems of linear equations in Laplacian matrices and their submatrices in nearly linear time. By “nearlylinear”, I mean time
The Laplacian Pyramid as a Compact Image Code
, 1983
"... We describe a technique for image encoding in which local operators of many scales but identical shape serve as the basis functions. The representation differs from established techniques in that the code elements are localized in spatial frequency as well as in space. Pixeltopixel correlations a ..."
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Cited by 1388 (12 self)
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is achieved by quantizing the difference image. These steps are then repeated to compress the lowpass image. Iteration of the process at appropriately expanded scales generates a pyramid data structure. The encoding process is equivalent to sampling the image with Laplacian operators of many scales. Thus
Fast linear iterations for distributed averaging.
 Systems & Control Letters,
, 2004
"... Abstract We consider the problem of finding a linear iteration that yields distributed averaging consensus over a network, i.e., that asymptotically computes the average of some initial values given at the nodes. When the iteration is assumed symmetric, the problem of finding the fastest converging ..."
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Cited by 433 (12 self)
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converging linear iteration can be cast as a semidefinite program, and therefore efficiently and globally solved. These optimal linear iterations are often substantially faster than several common heuristics that are based on the Laplacian of the associated graph. We show how problem structure can
A Fast LinearArithmetic Solver for DPLL(T)
, 2006
"... We present a new Simplexbased linear arithmetic solver that can be integrated efficiently in the DPLL(T) framework. The new solver improves over existing approaches by enabling fast backtracking, supporting a priori simplification to reduce the problem size, and providing an efficient form of the ..."
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Cited by 289 (13 self)
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We present a new Simplexbased linear arithmetic solver that can be integrated efficiently in the DPLL(T) framework. The new solver improves over existing approaches by enabling fast backtracking, supporting a priori simplification to reduce the problem size, and providing an efficient form
BerkMin: a fast and robust satsolver
, 2002
"... We describe a SATsolver, BerkMin, that inherits such features of GRASP, SATO, and Chaff as clause recording, fast BCP, restarts, and conflict clause “aging”. At the same time BerkMin introduces a new decision making procedure and a new method of clause database management. We experimentally compare ..."
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Cited by 284 (5 self)
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We describe a SATsolver, BerkMin, that inherits such features of GRASP, SATO, and Chaff as clause recording, fast BCP, restarts, and conflict clause “aging”. At the same time BerkMin introduces a new decision making procedure and a new method of clause database management. We experimentally
Automated Whitebox Fuzz Testing
"... Fuzz testing is an effective technique for finding security vulnerabilities in software. Traditionally, fuzz testing tools apply random mutations to wellformed inputs of a program and test the resulting values. We present an alternative whitebox fuzz testing approach inspired by recent advances in ..."
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Cited by 311 (25 self)
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and solved with a constraint solver, producing new inputs that exercise different control paths in the program. This process is repeated with the help of a codecoverage maximizing heuristic designed to find defects as fast as possible. We have implemented this algorithm in SAGE (Scalable, Automated, Guided
Lean algebraic multigrid (LAMG): Fast graph Laplacian linear solver
 ARXIV EPRINTS
"... Laplacian matrices of graphs arise in largescale computational applications such as semisupervised machine learning; spectral clustering of images, genetic data and web pages; transportation network flows; electrical resistor circuits; and elliptic partial differential equations discretized on un ..."
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Cited by 8 (0 self)
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on unstructured grids with finite elements. A Lean Algebraic Multigrid (LAMG) solver of the symmetric linear system Ax = b is presented, where A is a graph Laplacian. LAMG’s run time and storage are empirically demonstrated to scale linearly with the number of edges. LAMG consists of a setup phase during which a
Fast maximum margin matrix factorization for collaborative prediction
 In Proceedings of the 22nd International Conference on Machine Learning (ICML
, 2005
"... Maximum Margin Matrix Factorization (MMMF) was recently suggested (Srebro et al., 2005) as a convex, infinite dimensional alternative to lowrank approximations and standard factor models. MMMF can be formulated as a semidefinite programming (SDP) and learned using standard SDP solvers. However, cu ..."
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Cited by 248 (6 self)
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Maximum Margin Matrix Factorization (MMMF) was recently suggested (Srebro et al., 2005) as a convex, infinite dimensional alternative to lowrank approximations and standard factor models. MMMF can be formulated as a semidefinite programming (SDP) and learned using standard SDP solvers. However
Sixth Cargese Workshop on Combinatorial Optimization September 16, 2015 Lecture 2: Matrix Chernoff bounds
"... The purpose of my second and third lectures is to discuss spectral sparsifiers, which are the second key ingredient in most of the fast Laplacian solvers. In this lecture we will discuss concentration bounds for sums of random matrices, which are an important technical tool underlying the simplest s ..."
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The purpose of my second and third lectures is to discuss spectral sparsifiers, which are the second key ingredient in most of the fast Laplacian solvers. In this lecture we will discuss concentration bounds for sums of random matrices, which are an important technical tool underlying the simplest
A Fast PseudoBoolean Constraint Solver
, 2003
"... Linear PseudoBoolean (LPB) constraints denote inequalities between arithmetic sums of weighted Boolean functions and provide a significant extension of the modeling power of purely propositional constraints. They can be used to compactly describe many discrete EDA problems with constraints on linea ..."
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Cited by 115 (1 self)
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optimization method for many problems in logic and physical synthesis. In this paper we review how recent advances in satisfiability (SAT) search can be extended for pseudoBoolean constraints and describe a new LPB solver that is based on generalized constraint propagation and conflictbased learning. We
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