Generalized Graph Laplacian Research Articles

Handling information loss of graph convolutional networks in collaborative filtering

Collaborative filtering (CF) methods based on graph convolutional network (GCN) and autoencoder (AE) achieve outstanding performance. But the GCN-based CF methods suffer from information loss problems, which are caused by information lossy initialization and using low-order Chebyshev Polynomial to fit the graph convolution kernel. And the AE-based CF methods obtain the prediction results by reconstructing the user-item interaction matrix, which does not conduct deep excavation of the behavior patterns, resulting in the limited-expression ability.To solve the above problems, we propose Variational AutoEncoder-Enhanced Graph Convolutional Network (VE-GCN) for CF. Specifically, we use a variational autoencoder (VAE) to compress the interactive behavior patterns as the prior information of GCN to achieve sufficient learning, thus alleviating the information lossy initialization problem. And then the generalized graph Laplacian convolution kernel is proposed in GCN to handle the high-frequency information loss problem caused by Chebyshev Polynomial fitting in the GCN-based CF. To the best of our knowledge, VE-GCN is a feasible method to handle the information loss problems mentioned above in GCN-based CF for the first time. Meanwhile, the structure of GCN is optimized by removing redundant feature transformation and nonlinear activation function, and using DenseGCN to complete multi-level information interaction. Experiments on four real-world datasets show that the VE-GCN achieves state-of-the-art performance.

Generalized Graph Laplacian Based Anomaly Detection for Spatiotemporal MicroPMU Data

This letter develops a novel anomaly detection method using the generalized graph Laplacian (GGL) matrix to visualize the spatiotemporal relationship of distribution-level phasor measurement unit ( $\boldsymbol{\mu }$ PMU) data. The $\boldsymbol{\mu }$ PMU data in a specific time horizon are segregated into multiple segments. An optimization problem formulated as a Lagrangian function is utilized to estimate the GGL matrix. During the iterative process, an optimal update is constituted as a quadratic program problem. To perform the $\boldsymbol{\mu }$ PMU-based spatiotemporal analysis, normalized diagonal elements of GGL matrix are proposed as a quantitative metric. The effectiveness of the developed method is demonstrated through real-world $\boldsymbol{\mu }$ PMU measurements gathered from test feeders in Riverside, CA, USA.

Graph Laplacian Regularization with Sparse Coding for Image Restoration and Representation

Sparse coding is widely used in image denoising, deblurring, clustering, and classification. However, most existing approaches to sparse coding failed to consider the fact that high dimensional data naturally reside on geometrical structure of the data space. It has been shown that geometric information of the data is important for both inversion and discrimination. In this paper, we proposed a generalized framework for image restoration and representation by combining sparse coding and graph based algorithms. In image denoising and deblurring problems, an image is first decomposed into cartoon layer (piecewise-smooth contents) and texture layer (textures and sharp edges) using morphological component analysis (MCA); then optimal graph Laplacian regularizer (OGLR) algorithm and simultaneous sparse coding with Gaussian scale mixture prior (SSC-GSM) algorithm are applied to cartoon layer and texture layer, respectively; final restored image is generated by aggregating the outcomes from two algorithms. The proposed hybrid image restoration algorithm outperforms state-of-the-art image denoising algorithms, such as BM3D on natural images, measured in PSNR, and performs comparatively in image deblurring. In image clustering and classification problems, we convert our generalized framework into a novel dual graph regularized sparse coding method to transform the nonlinear data space and feature space into linear space, two efficient optimization algorithms are provided for the numerical implementation. The experimental results show that our generalized graph Laplacian and sparse coding framework performs competitively with popular denoising, deblurring, clustering, and classification methods.

Duality and nonlinear graph Laplacians

In this paper we show that a recent nearly linear time algorithm for solving a system of equations arising from a graph Laplacian can be extended to a large class of nonlinear systems of equations, based on a nonlinear generalization of the graph Laplacian. This result follows from a nonlinear generalization of the notable duality between graph Laplacians and electrical flows. Beyond providing a fast algorithm for a class of nonlinear sets of equations, our work highlights the robustness of spectral graph theory and suggests new directions for research in spectral graph theory.Specifically, we present an iterative algorithm for solving a class of nonlinear system of equations arising from a nonlinear generalization of the graph Laplacian in O˜(k2mlog⁡(kn/ϵ)) iterations, where k is a measure of nonlinearity, n is the number of variables, m is the number of nonzero entries in the generalized graph Laplacian L, ϵ is the solution accuracy and O˜() neglects (non-leading) logarithmic terms. This algorithm is a natural nonlinear extension of the one in Kelner et al. (2013), which solves a linear Laplacian system of equations in nearly linear time. Unlike the linear case, in the nonlinear case each iteration takes O˜(n) time so the total running time is O˜(k2mnlog⁡(kn/ϵ)). For sparse graphs with m=O(n) and fixed k this nonlinear algorithm is O˜(n2log⁡(n/ϵ)) which is slightly faster than standard methods for solving linear equations, which require approximately O(n2.38) time.

Two types of discrete Sobolev inequalities on a weighted Toeplitz graph

In this paper, two types of discrete Sobolev inequalities that correspond to the generalized graph Laplacian A on a weighted Toeplitz graph are obtained. The sharp constants C0(a) and C0 are calculated using the Green matrix G(a)=(A+aI)−1(0<a<∞) and pseudo-Green matrix G⁎=A† (Penrose–Moore generalized inverse matrix of A). The sharp constants are expressed as reciprocals of the harmonic mean corresponding to eigenvalues of each matrix A+aI and A except an eigenvalue 0.

A generalization of the graph Laplacian with application to a distributed consensus algorithm

Abstract In order to describe the interconnection among agents with multi-dimensional states, we generalize the notion of a graph Laplacian by extending the adjacency weights (or weighted interconnection coefficients) from scalars to matrices. More precisely, we use positive definite matrices to denote full multi-dimensional interconnections, while using nonnegative definite matrices to denote partial multi-dimensional interconnections. We prove that the generalized graph Laplacian inherits the spectral properties of the graph Laplacian. As an application, we use the generalized graph Laplacian to establish a distributed consensus algorithm for agents described by multi-dimensional integrators.

Nodal domain theorems and bipartite subgraphs

The Discrete Nodal Domain Theorem states that an eigenfunction of the k-th largesteigenvalue of a generalized graph Laplacian has at most k (weak) nodal domains. The number ofstrong nodal domains is shown not to exceed the size of a maximal induced bipartite subgraph andthat this bound is sharp for generalized graph Laplacians. Similarly, the number of weak nodaldomains is bounded by the size of a maximal bipartite minor.