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Register a new userFractional spectral graph wavelets and their applications
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Added by
Jiasong Wu
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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One of the key challenges in the area of signal processing on graphs is to design transforms and dictionaries methods to identify and exploit structure in signals on weighted graphs. In this paper, we first generalize graph Fourier transform (GFT) to graph fractional Fourier transform (GFRFT), which is then used to define a novel transform named spectral graph fractional wavelet transform (SGFRWT), which is a generalized and extended version of spectral graph wavelet transform (SGWT). A fast algorithm for SGFRWT is also derived and implemented based on Fourier series approximation. The potential applications of SGFRWT are also presented.
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Spectral graph convolutional networks (SGCNs) have been attracting increasing attention in graph representation learning partly due to their interpretability through the prism of the established graph signal processing framework. However, existing SGCNs are limited in implementing graph convolutions with rigid transforms that could not adapt to signals residing on graphs and tasks at hand. In this paper, we propose a novel class of spectral graph convolutional networks that implement graph convolutions with adaptive graph wavelets. Specifically, the adaptive graph wavelets are learned with neural network-parameterized lifting structures, where structure-aware attention-based lifting operations are developed to jointly consider graph structures and node features. We propose to lift based on diffusion wavelets to alleviate the structural information loss induced by partitioning non-bipartite graphs. By design, the locality and sparsity of the resulting wavelet transform as well as the scalability of the lifting structure for large and varying-size graphs are guaranteed. We further derive a soft-thresholding filtering operation by learning sparse graph representations in terms of the learned wavelets, which improves the scalability and interpretablity, and yield a localized, efficient and scalable spectral graph convolution. To ensure that the learned graph representations are invariant to node permutations, a layer is employed at the input of the networks to reorder the nodes according to their local topology information. We evaluate the proposed networks in both node-level and graph-level representation learning tasks on benchmark citation and bioinformatics graph datasets. Extensive experiments demonstrate the superiority of the proposed networks over existing SGCNs in terms of accuracy, efficiency and scalability.
These are notes on the method of normalized graph cuts and its applications to graph clustering. I provide a fairly thorough treatment of this deeply original method due to Shi and Malik, including complete proofs. I include the necessary background on graphs and graph Laplacians. I then explain in detail how the eigenvectors of the graph Laplacian can be used to draw a graph. This is an attractive application of graph Laplacians. The main thrust of this paper is the method of normalized cuts. I give a detailed account for K = 2 clusters, and also for K > 2 clusters, based on the work of Yu and Shi. Three points that do not appear to have been clearly articulated before are elaborated: 1. The solutions of the main optimization problem should be viewed as tuples in the K-fold cartesian product of projective space RP^{N-1}. 2. When K > 2, the solutions of the relaxed problem should be viewed as elements of the Grassmannian G(K,N). 3. Two possible Riemannian distances are available to compare the closeness of solutions: (a) The distance on (RP^{N-1})^K. (b) The distance on the Grassmannian. I also clarify what should be the necessary and sufficient conditions for a matrix to represent a partition of the vertices of a graph to be clustered.
(Draft 3) A generalized differential operator on the real line is defined by means of a limiting process. These generalized derivatives include, as a special case, the classical derivative and current studies of fractional differential operators. All such operators satisfy properties such as the sum, product/quotient rules, chain rule, etc. We study a Sturm-Liouville eigenvalue problem with generalized derivatives and show that the general case is actually a consequence of standard Sturm-Liouville Theory. As an application of the developments herein we find the general solution of a generalized harmonic oscillator. We also consider the classical problem of a planar motion under a central force and show that the general solution of this problem is still generically an ellipse, and that this result is true independently of the choice of the generalized derivatives being used modulo a time shift. The previous result on the generic nature of phase plane orbits is extended to the classical gravitational n-body problem of Newton to show that the global nature of these orbits is independent of the choice of the generalized derivatives being used in defining the force law (modulo a time shift). Finally, restricting the generalized derivatives to a special class of fractional derivatives, we consider the question of motion under gravity with and without resistance and arrive at a new notion of time that depends on the fractional parameter. The results herein are meant to clarify and extend many known results in the literature and intended to show the limitations and use of generalized derivatives and corresponding fractional derivatives.
Knowledge Graph embedding provides a versatile technique for representing knowledge. These techniques can be used in a variety of applications such as completion of knowledge graph to predict missing information, recommender systems, question answering, query expansion, etc. The information embedded in Knowledge graph though being structured is challenging to consume in a real-world application. Knowledge graph embedding enables the real-world application to consume information to improve performance. Knowledge graph embedding is an active research area. Most of the embedding methods focus on structure-based information. Recent research has extended the boundary to include text-based information and image-based information in entity embedding. Efforts have been made to enhance the representation with context information. This paper introduces growth in the field of KG embedding from simple translation-based models to enrichment-based models. This paper includes the utility of the Knowledge graph in real-world applications.
In this paper, we present a new way to associate a finitely summable spectral triple to a higher-rank graph $Lambda$, via the infinite path space $Lambda^infty$ of $Lambda$. Moreover, we prove that this spectral triple has a close connection to the wavelet decomposition of $Lambda^infty$ which was introduced by Farsi, Gillaspy, Kang, and Packer in 2015. We first introduce the concept of stationary $k$-Bratteli diagrams, in order to associate a family of ultrametric Cantor sets, and their associated Pearson-Bellissard spectral triples, to a finite, strongly connected higher-rank graph $Lambda$. We then study the zeta function, abscissa of convergence, and Dixmier trace associated to the Pearson-Bellissard spectral triples of these Cantor sets, and show these spectral triples are $zeta$-regular in the sense of Pearson and Bellissard. We obtain an integral formula for the Dixmier trace given by integration against a measure $mu$, and show that $mu$ is a rescaled version of the measure $M$ on $Lambda^infty$ which was introduced by an Huef, Laca, Raeburn, and Sims. Finally, we investigate the eigenspaces of a family of Laplace-Beltrami operators associated to the Dirichlet forms of the spectral triples. We show that these eigenspaces refine the wavelet decomposition of $L^2(Lambda^infty, M)$ which was constructed by Farsi et al.
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