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Wavelets on Graphs via Spectral Graph Theory

TLDR
In this paper, the spectral graph wavelet operator is defined based on spectral decomposition of the discrete graph Laplacian, and a wavelet generating kernel and a scale parameter are used to localize this operator to an indicator function.
Abstract
We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. Our approach is based on defining scaling using the the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian $\L$. Given a wavelet generating kernel $g$ and a scale parameter $t$, we define the scaled wavelet operator $T_g^t = g(t\L)$. The spectral graph wavelets are then formed by localizing this operator by applying it to an indicator function. Subject to an admissibility condition on $g$, this procedure defines an invertible transform. We explore the localization properties of the wavelets in the limit of fine scales. Additionally, we present a fast Chebyshev polynomial approximation algorithm for computing the transform that avoids the need for diagonalizing $\L$. We highlight potential applications of the transform through examples of wavelets on graphs corresponding to a variety of different problem domains.

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Citations
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Proceedings ArticleDOI

Detection of Internet Traffic Anomalies Using Sparse Laplacian Component Analysis

TL;DR: This work proposes a regression framework to compute Laplacian components followed by its application in anomaly detection and demonstrates that the proposed model can correctly uncover the essential low-dimensional principal subspace containing the normal Internet traffic and achieve outstanding detection performance.
Posted Content

Community-preserving Graph Convolutions for Structural and Functional Joint Embedding of Brain Networks

TL;DR: Wang et al. as discussed by the authors proposed a framework of Siamese community-preserving graph convolutional network (SCP-GCN) to learn the structural and functional joint embedding of brain networks.
Proceedings ArticleDOI

Multiresolution analysis of incomplete rankings with applications to prediction

TL;DR: A new representation of preference data when they come under the form of incomplete rankings, that is to say ordinal preferences on small subsets of items is promoted and it is shown that it is statistically consistent and that it can be computed at a reasonable cost given the complexity of the original data.
Proceedings ArticleDOI

Image colorization based on ADMM with fast singular value thresholding by Chebyshev polynomial approximation

TL;DR: This paper reduces the computational cost of singular value thresholding by using Chebyshev polynomial approximation (CPA), and replaces the optimization method used in the image colorization method by alternating direction method of multipliers, which further accelerates the computation.
Book ChapterDOI

Analysis of Framelet Transforms on a Simplex

TL;DR: The framelet transforms—decomposition and reconstruction of the coefficients for framelets of a function on T2 are given and it is proved that the reconstruction is exact when the framelets are tight.
References
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Book

A wavelet tour of signal processing

TL;DR: An introduction to a Transient World and an Approximation Tour of Wavelet Packet and Local Cosine Bases.
Proceedings ArticleDOI

Object recognition from local scale-invariant features

TL;DR: Experimental results show that robust object recognition can be achieved in cluttered partially occluded images with a computation time of under 2 seconds.
Book

Ten lectures on wavelets

TL;DR: This paper presents a meta-analyses of the wavelet transforms of Coxeter’s inequality and its applications to multiresolutional analysis and orthonormal bases.
Book

Functional analysis

Walter Rudin
Journal ArticleDOI

Ten Lectures on Wavelets

TL;DR: In this article, the regularity of compactly supported wavelets and symmetry of wavelet bases are discussed. But the authors focus on the orthonormal bases of wavelets, rather than the continuous wavelet transform.
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