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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TL;DR: The analysis reveals the limitations of the standard degree-based normalization method in that the resulting normalization factors can vary significantly within each connected component with the same class label, which may cause inferior generalization performance.
Abstract: This paper investigates the effect of Laplacian normalization in graph-based semi-supervised learning. To this end, we consider multi-class transductive learning on graphs with Laplacian regularization. Generalization bounds are derived using geometric properties of the graph. Specifically, by introducing a definition of graph cut from learning theory, we obtain generalization bounds that depend on the Laplacian regularizer. We then use this analysis to better understand the role of graph Laplacian matrix normalization. Under assumptions that the cut is small, we derive near-optimal normalization factors by approximately minimizing the generalization bounds. The analysis reveals the limitations of the standard degree-based normalization method in that the resulting normalization factors can vary significantly within each connected component with the same class label, which may cause inferior generalization performance. Our theory also suggests a remedy that does not suffer from this problem. Experiments confirm the superiority of the normalization scheme motivated by learning theory on artificial and real-world data sets.
76 citations
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01 Jan 2014TL;DR: Techniques from spectral graph theory are applied to analyze repeated patterns in musical recordings and produce a low-dimensional encoding of repetition structure, and exposes the hierarchical relationships among structural components at differing levels of granularity.
Abstract: Many approaches to analyzing the structure of a musical recording involve detecting sequential patterns within a selfsimilarity matrix derived from time-series features. Such patterns ideally capture repeated sequences, which then form the building blocks of large-scale structure. In this work, techniques from spectral graph theory are applied to analyze repeated patterns in musical recordings. The proposed method produces a low-dimensional encoding of repetition structure, and exposes the hierarchical relationships among structural components at differing levels of granularity. Finally, we demonstrate how to apply the proposed method to the task of music segmentation.
76 citations
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TL;DR: This paper shows that the there are topological constraints on the index of the Laplacian matrix related to the dimension of a certain homology group, which gives bounds on the number of positive and negative eigenvalues.
Abstract: Many applied problems can be posed as a dynamical system defined on a network with attractive and repulsive interactions. Examples include synchronization of nonlinear oscillator networks; the behavior of groups, or cliques, in social networks; and the study of optimal convergence for consensus algorithm. It is important to determine the index of a matrix, i.e., the number of positive and negative eigenvalues, and the dimension of the kernel. In this paper we consider the common examples where the matrix takes the form of a signed graph Laplacian. We show that the there are topological constraints on the index of the Laplacian matrix related to the dimension of a certain homology group. When the homology group is trivial, the index of the operator is determined only by the topology of the network and is independent of the strengths of the interactions. In general, these constraints give bounds on the number of positive and negative eigenvalues, with the dimension of the homology group counting the number ...
76 citations
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24 Oct 1984TL;DR: In this article, the eigenvalues of the adjacency matrix of a graph and its expansion properties were used to construct explicitly n-superconcentrators with 157.4 n edges, much less than the previous most economical construction.
Abstract: Explicit construction of families of linear expanders and superconcentrators is relevant to theoretical computer science in several ways. There is essentially only one known explicit construction. Here we show a correspondence between the eigenvalues of the adjacency matrix of a graph and its expansion properties, and combine it with results on Group Representations to obtain many new examples of families of linear expanders. We also obtain better expanders than those previously known and use them to construct explicitly n-superconcentrators with 157.4 n edges, much less than the previous most economical construction.
75 citations
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TL;DR: In this paper, general Berry-Esseen bounds are developed for the exponential distri- bution using Stein's method and a new concentration inequality approach, and a sharp error term for Hora's result that the spectrum of the Johnson graph has an exponential limit.
Abstract: General Berry-Esseen bounds are developed for the exponential distri- bution using Stein's method and a new concentration inequality approach. As an application, a sharp error term is obtained for Hora's result that the spectrum of the Johnson graph has an exponential limit.
74 citations