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Approximate Gaussian Elimination for Laplacians: Fast, Sparse, and Simple [article]

Rasmus Kyng, Sushant Sachdeva
2016 arXiv   pre-print
We show how to perform sparse approximate Gaussian elimination for Laplacian matrices.  ...  We present a simple, nearly linear time algorithm that approximates a Laplacian by a matrix with a sparse Cholesky factorization, the version of Gaussian elimination for symmetric matrices.  ...  Acknowledgements We thank Daniel Spielman for suggesting this project and for helpful comments and discussions.  ... 
arXiv:1605.02353v1 fatcat:zocjrggnnzc2rhnue3iuomvzsy

Graph Sparsification Approaches for Laplacian Smoothing

Veeranjaneyulu Sadhanala, Yu-Xiang Wang, Ryan J. Tibshirani
2016 International Conference on Artificial Intelligence and Statistics  
We provide strong theoretical and experimental results, demonstrating that sparsification before estimation can give statistically sensible solutions, with significant computational savings.  ...  performed according to the structure of a large, dense graph G, we consider fitting the statistical estimate using a sparsified surrogate graph G, which shares the vertices of G but has far fewer edges, and  ...  We thank Alex Smola for his general help and guidance, and Ioannis Koutis and Shen Chen Xu for help with their sparsifier code.  ... 
dblp:conf/aistats/SadhanalaWT16 fatcat:ikzp6ravi5hfhi57mswh2obq7e

Fast Image Deconvolution using Hyper-Laplacian Priors

Dilip Krishnan, Rob Fergus
2009 Neural Information Processing Systems  
However, the use of sparse distributions makes the problem non-convex and impractically slow to solve for multi-megapixel images.  ...  Alternatively, for two specific values of α, 1/2 and 2/3 an analytic solution can be found, by finding the roots of a cubic and quartic polynomial, respectively.  ...  Other types of sparse image prior include: Gaussian Scale Mixtures (GSM) [21] , which have been used for image deblurring [3] and denoising [14] and student-T distributions for denoising [25, 16]  ... 
dblp:conf/nips/KrishnanF09 fatcat:7gbrly4irjhgznxch7g6briwcm

Laplacian Eigenimages in Discrete Scale Space [chapter]

Martin Tschirsich, Arjan Kuijper
2012 Lecture Notes in Computer Science  
In this paper, relevant and promising properties of the discrete diffusion equation and the eigenvalue decomposition of its Laplacian kernel are discussed and a fast and robust sampling method is proposed  ...  Linear or Gaussian scale space is a well known multi-scale representation for continuous signals. However, implementational issues arise, caused by discretization and quantization errors.  ...  The next section then provides a simple solution of the discretized diffusion equation, enabling for fast and accurate sampling of L.  ... 
doi:10.1007/978-3-642-34166-3_18 fatcat:ub4k45c5ingbbg6ytfe3s6eqj4

On Fast Computation of Directed Graph Laplacian Pseudo-Inverse [article]

Daniel Boley
2020 arXiv   pre-print
The Laplacian matrix and its pseudo-inverse for a strongly connected directed graph is fundamental in computing many properties of a directed graph.  ...  The approach is based on "off-the-shelf" iterative methods for which global linear convergence is guaranteed, without recourse to any matrix elimination algorithm.  ...  Acknowledgements This research was supported in part by NSF grants 1835530 and 1922512.  ... 
arXiv:2002.12773v3 fatcat:jgeucg7cirgsvj5pfcnshatyfy

Algorithms, Graph Theory, and Linear Equations in Laplacian Matrices

Daniel A. Spielman
2011 Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)  
In this talk we survey recent progress on the design of provably fast algorithms for solving linear equations in the Laplacian matrices of graphs.  ...  These are all connected by a definition of what it means for one graph to approximate another.  ...  Approximation by Sparse Graphs Sparsification is the process of approximating a given graph G by a sparse graph H.  ... 
doi:10.1142/9789814324359_0164 fatcat:mqplntp7hbf33no4djeyaduqzy

A Posteriori Error Estimates for Solving Graph Laplacians [article]

Xiaozhe Hu, Kaiyi Wu, Ludmil T. Zikatanov
2021 arXiv   pre-print
In this paper, we study a posteriori error estimators which aid multilevel iterative solvers for linear systems with graph Laplacians.  ...  As we show, such an estimator has a nearly-linear computational complexity for sparse graphs under certain assumptions.  ...  The first step can be done in linear time with Gaussian elimination with special ordering [51, 45] .  ... 
arXiv:2007.00189v2 fatcat:swdhlqgypnewhn35dzjgap6vwm

Sparse matrix factorizations for fast linear solvers with application to Laplacian systems [article]

Michael T. Schaub, Maguy Trefois, Paul Van Dooren, Jean-Charles Delvenne
2016 arXiv   pre-print
We thereby can connect our results to recently proposed nearly-linear time solvers for Laplacian systems, which emerge here as a particular application of our sparse matrix factorization.  ...  In this paper, we propose to interpolate between these two extremes, and show how to perform cheap iterations along non-sparse search directions, provided that these directions can be extracted from a  ...  First, there are direct methods [7] like Cholesky factorization or Gaussian elimination.  ... 
arXiv:1605.09148v2 fatcat:h4cexh5hmjhldbnwwfroimz5o4

Spectral Sparse Representation for Clustering: Evolved from PCA, K-means, Laplacian Eigenmap, and Ratio Cut [article]

Zhenfang Hu, Gang Pan, Yueming Wang, Zhaohui Wu
2017 arXiv   pre-print
The methods include PCA, K-means, Laplacian eigenmap (LE), ratio cut (Rcut), and a new sparse representation method developed by us, called spectral sparse representation (SSR).  ...  For its inherent relation to cluster analysis, the codes of SSR can be directly used for clustering.  ...  SPCArt is a sparse PCA algorithm designed to solve sparse loadings. It finds a rotation matrix and sparse loadings such that the sparse loadings approximate the PCA loadings after the rotation.  ... 
arXiv:1403.6290v4 fatcat:ftmibom7frabhhodj6abamz47m

Sparsified Cholesky and Multigrid Solvers for Connection Laplacians [article]

Rasmus Kyng, Yin Tat Lee, Richard Peng, Sushant Sachdeva, Daniel A. Spielman
2015 arXiv   pre-print
These algorithms accelerate Gaussian elimination by sparsifying the nonzero matrix entries created by the elimination process.  ...  We introduce the sparsified Cholesky and sparsified multigrid algorithms for solving systems of linear equations.  ...  The sparsified Cholesky algorithm accelerates Gaussian elimination by sparsifying the rows that are produced by elimination, thereby guaranteeing that the elimination will be fast.  ... 
arXiv:1512.01892v1 fatcat:uaarmuaqqfbezdo4a75n4jmjqa

Estimating Current-Flow Closeness Centrality with a Multigrid Laplacian Solver [article]

Elisabetta Bergamini, Michael Wegner, Dimitar Lukarski, Henning Meyerhenke
2020 arXiv   pre-print
To allow network analysis workflows to profit from a fast Laplacian solver, we provide an implementation of the LAMG multigrid solver in the NetworKit package, facilitating the computation of current-flow  ...  One field where Laplacian linear systems play a role is network analysis, e. g. for certain centrality measures that indicate if a node (or an edge) is important in the network.  ...  Acknowledgments This work was partially supported by German Research Foundation (DFG) grant ME3619/3-1 within Priority Programme 1736 Algorithms for Big Data.  ... 
arXiv:1607.02955v2 fatcat:2t6hbahoengjhkj64erstlghfm

Laplacian Eigenfunctions for Climate Analysis

Timothy DelSole, Michael K. Tippett
2015 Journal of Climate  
MATLAB and R codes for deriving Laplacian eigenfunctions are available upon request.  ...  This paper proposes a new method for representing data in a general domain on a sphere.  ...  We thank Dimitris Giannakis for insightful comments that led to significant clarifications in this paper.  ... 
doi:10.1175/jcli-d-15-0049.1 fatcat:l2o6zdulp5arlalyjxjkdgc7nm

The Surface Laplacian Technique in EEG: Theory and Methods [article]

Claudio Carvalhaes, J. Acacio de Barros
2014 arXiv   pre-print
Here we develop a set of discrete approximations for Laplacian estimates at peripheral electrodes and a possible solution to the problem of multiple-frame regularization.  ...  The latter eliminates all issues related to discretization and incorporates a regularization mechanism to reduce spatial noise, but at the cost of increasing mathematical and computational complexity.  ...  This method is conceptually simple and easy to implement, and for this reason still very popular.  ... 
arXiv:1406.0458v2 fatcat:xo24t6qrdbgrni4kl45m3qe6hu

Minor Sparsifiers and the Distributed Laplacian Paradigm [article]

Sebastian Forster and Gramoz Goranci and Yang P. Liu and Richard Peng and Xiaorui Sun and Mingquan Ye
2022 arXiv   pre-print
SICOMP'18] considered the approximate setting and works only for undirected graphs.  ...  We study distributed algorithms built around minor-based vertex sparsifiers, and give the first algorithm in the CONGEST model for solving linear systems in graph Laplacian matrices to high accuracy.  ...  This is because both of these sets may have size up to Ω( ), and passing that information to a single low degree vertex would incur too much communication.  ... 
arXiv:2012.15675v3 fatcat:bvoh5dhsirehpcx46vl54xokya

Efficient preconditioning of laplacian matrices for computer graphics

Dilip Krishnan, Raanan Fattal, Richard Szeliski
2013 ACM Transactions on Graphics  
By applying these operations before each elimination step and repeating the procedure recursively on the resulting smaller systems, we obtain a highly efficient multi-level preconditioning scheme with  ...  This speedup is achieved by the new method's ability to reduce the condition number of irregular Laplacian matrices as well as homogeneous systems.  ...  Acknowledgements We thank Alec Jacobson, Denis Zorin as well as the SIGGRAPH reviewers for useful suggestions and discussions, and Oren Livne for help with LAMG code and test data.  ... 
doi:10.1145/2461912.2461992 fatcat:zdth6wbvqvbrxozxklnw4zkowa
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