Topic
Spectral graph theory
About: Spectral graph theory is a research topic. Over the lifetime, 1334 publications have been published within this topic receiving 77373 citations.
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15 May 2006
TL;DR: This paper presents a novel approach for automatically obtaining consistent local maps from observations by considering the space sensed in each observation as a node of a graph with arcs representing the space overlap between observations.
Abstract: Recently, hybrid maps that combine metric and topological world information have been proposed as a powerful representation of mobile robot environments. Among others, these maps are of special interest for efficiently managing large-scale environments, and for accurate localization. For achieving that, local geometric maps are stored in the nodes of a graph-based global map. In this paper we present a novel approach for automatically obtaining those local maps from observations. The method considers the space sensed in each observation as a node of a graph with arcs representing the space overlap between observations. The recursive partition (cut) of this graph produces groups of strongly connected nodes from which consistent local maps for accurate localization are derived. The proposed partition technique is well-grounded in the spectral graph theory of, and it is formulated for any type of sensor observation. We depict an implementation for grouping 2D laser scans, and show experimental results with real data that demonstrate the performance of the method
44 citations
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26 Jun 2015TL;DR: A toolset based on spectral sparsification for a family of fundamental problems involving Gaussian sampling, matrix functionals, and reversible Markov chains is developed, which is expected to strengthen the connection between machine learning and spectral graph theory, two of the most active fields in understanding large data and networks.
Abstract: Motivated by a sampling problem basic to computational statistical inference, we develop a toolset based on spectral sparsification for a family of fundamental problems involving Gaussian sampling, matrix functionals, and reversible Markov chains. Drawing on the connection between Gaussian graphical models and the recent breakthroughs in spectral graph theory, we give the first nearly linear time algorithm for the following basic matrix problem: Given an n× n Laplacian matrix M and a constant −1 ≤ p ≤ 1, provide efficient access to a sparse n× n linear operator C such that M ≈ CC>, where ≈ denotes spectral similarity. When p is set to −1, this gives the first parallel sampling algorithm that is essentially optimal both in total work and randomness for Gaussian random fields with symmetric diagonally dominant (SDD) precision matrices. It only requires nearly linear work and 2n i.i.d. random univariate Gaussian samples to generate an n-dimensional i.i.d. Gaussian random sample in polylogarithmic depth. The key ingredient of our approach is an integration of spectral sparsification with multilevel method: Our algorithms are based on factoring M into a product of well-conditioned matrices, then introducing powers and replacing dense matrices with sparse approximations. We give two sparsification methods for this approach that may be of independent interest. The first invokes Maclaurin series on the factors, while the second builds on our new nearly linear time spectral sparsification algorithm for random-walk matrix polynomials. We expect these algorithmic advances will also help to strengthen the connection between machine learning and spectral graph theory, two of the most active fields in understanding large data and networks.
44 citations
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TL;DR: The experimental results on a variety of 3D models demonstrate the effectiveness of the proposed hashing technique in terms of robustness against the most common attacks including Gaussian noise, mesh smoothing, mesh compression, scaling, rotation as well as combinations of these attacks.
Abstract: Identification and authentication of multimedia content has become one of the most important aspects of multimedia security. In this paper, we present a hashing technique for 3D models using spectral graph theory and entropic spanning trees. The main idea is to partition a 3D triangle mesh into an ensemble of sub-meshes, then apply eigen-decomposition to the Laplace-Beltrami matrix of each sub-mesh, followed by computing the hash value of each sub-mesh. This hash value is defined in terms of spectral coefficients and Tsallis entropy estimate. The experimental results on a variety of 3D models demonstrate the effectiveness of the proposed technique in terms of robustness against the most common attacks including Gaussian noise, mesh smoothing, mesh compression, scaling, rotation as well as combinations of these attacks.
44 citations
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TL;DR: The results demonstrate that the MSD approach is able to outperform the traditional methods and help detect AD at an early stage, probably due to the success of exploiting the manifold structure of the data.
43 citations
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TL;DR: In this article, the Laplacian Estrada index of the graph G, LEE=LEE(G)= ∑n/i=1 e(µi - 2m/n).
Abstract: Let G be a graph of order n. Let λ1 , λ2 , . . . , λn be the eigenvalues of the adjacency matrix of G, and let µ1 , µ2 , . . . , µn be the eigenvalues of the Laplacian matrix of G. Much studied Estrada index of the graph G is defined n as EE = EE(G)= ∑n/i=1 eλi . We define and investigate the Laplacian Estrada index of the graph G, LEE=LEE(G)= ∑n/i=1 e(µi - 2m/n). Bounds for LEE are obtained, as well as some relations between LEE and graph Laplacian energy.
43 citations