Spectrum-Adapted Polynomial Approximation for Matrix Functions with Applications in Graph Signal Processing (original) (raw)

Localized Fourier Analysis for Graph Signal Processing

arXiv: Signal Processing, 2020

We propose a new point of view in the study of Fourier analysis on graphs, taking advantage of localization in the Fourier domain. For a signal fff on vertices of a weighted graph mathcalG\mathcal{G}mathcalG with Laplacian matrix mathcalL\mathcal{L}mathcalL, standard Fourier analysis of fff relies on the study of functions g(mathcalL)fg(\mathcal{L})fg(mathcalL)f for some filters ggg on ImathcalLI_\mathcal{L}ImathcalL, the smallest interval containing the Laplacian spectrum rmsp(mathcalL)subsetImathcalL{\rm sp}(\mathcal{L}) \subset I_\mathcal{L}rmsp(mathcalL)subsetImathcalL. We show that for carefully chosen partitions ImathcalL=sqcup1leqkleqKIkI_\mathcal{L} = \sqcup_{1\leq k\leq K} I_kImathcalL=sqcup1leqkleqKIk ($I_k \subset I_\mathcal{L}$), there are many advantages in understanding the collection (g(mathcalLIk)f)1leqkleqK(g(\mathcal{L}_{I_k})f)_{1\leq k\leq K}(g(mathcalLIk)f)1leqkleqK instead of g(mathcalL)fg(\mathcal{L})fg(mathcalL)f directly, where mathcalLI\mathcal{L}_ImathcalLI is the projected matrix PI(mathcalL)mathcalLP_I(\mathcal{L})\mathcal{L}PI(mathcalL)mathcalL. First, the partition provides a convenient modelling for the study of theoretical properties of Fourier analysis and allows for new results in graph signal analysis (\emph{e.g.} noise level estimation,...

On Local Distributions in Graph Signal Processing

ArXiv, 2022

Graph filtering is the cornerstone operation in graph signal processing (GSP). Thus, understanding it is key in developing potent GSP methods. Graph filters are local and distributed linear operations, whose output depends only on the local neighborhood of each node. Moreover, a graph filter’s output can be computed separately at each node by carrying out repeated exchanges with immediate neighbors. Graph filters can be compactly written as polynomials of a graph shift operator (typically, a sparse matrix description of the graph). This has led to relating the properties of the filters with the spectral properties of the corresponding matrix – which encodes global structure of the graph. In this work, we propose a framework that relies solely on the local distribution of the neighborhoods of a graph. The crux of this approach is to describe graphs and graph signals in terms of a measurable space of rooted balls. Leveraging this, we are able to seamlessly compare graphs of different ...

Complex Basis For Spectral Analysis of Graph Signals

International Journal of Mathematics Trends and Technology, 2020

The Signal Processing on Graph (SPG) is an emerging field of research aiming to develop accurate methods for big data analysis by combining graph theory and classical signal processing methods. One key method in signal processing on graph is the so-called Graph Fourier Transform (GFT) which is a generalization of the Classical Fourier Transform (defined for data lying on regular domains :1D for times series or 2D for images) to data lying on networks. Those network data are viewed like a set of interrelated data points lying on a graph whose graph vertices map the data points and graph links encode the relationship between data. In the classical framework, the Fourier transform is a linear operator that performs the mapping of a vector from its initial representation domain to the frequency domain through the Fourier matrix which is an orthonormal basis formed by complex exponential vectors constructed from powers of the complex number. Those vectors are of a key importance in the properties of the transform and its applications. However, for each graph Fourier transform proposed in the literature, although its graph Fourier matrix is orthonormal, its vectors are not complex as in the classical framework, limiting the extension and the use of some useful properties of the classical Fourier transform to the graph signals framework. In this work, we present a method to define a complex orthonormal basis for the graph Fourier transform that allows to perform spectral analysis for graph signals in the frequency domain. The graph Fourier basis we defined is identical to the Fourier basis when applied to graph signals defined on a regular domain. We applied the proposed method successfully to signal detection on an irregularly sampled sensor network.

Graph Signal Processing -- Part I: Graphs, Graph Spectra, and Spectral Clustering

arXiv (Cornell University), 2019

The area of Data Analytics on graphs promises a paradigm shift as we approach information processing of classes of data, which are typically acquired on irregular but structured domains (social networks, various ad-hoc sensor networks). Yet, despite its long history, current approaches mostly focus on the optimization of graphs themselves, rather than on directly inferring learning strategies, such as detection, estimation, statistical and probabilistic inference, clustering and separation from signals and data acquired on graphs. To fill this void, we first revisit graph topologies from a Data Analytics point of view, and establish a taxonomy of graph networks through a linear algebraic formalism of graph topology (vertices, connections, directivity). This serves as a basis for spectral analysis of graphs, whereby the eigenvalues and eigenvectors of graph Laplacian and adjacency matrices are shown to convey physical meaning related to both graph topology and higher-order graph properties, such as cuts, walks, paths, and neighborhoods. Through a number of carefully chosen examples, we demonstrate that the isomorphic nature of graphs enables the basic properties and descriptors to be preserved throughout the data analytics process, even in the case of reordering of graph vertices, where classical approaches fail. Next, to illustrate estimation strategies performed on graph signals, spectral analysis of graphs is introduced through eigenanalysis of mathematical descriptors of graphs and in a generic way. Finally, a framework for vertex clustering and graph segmentation is established based on graph spectral representation (eigenanalysis) which illustrates the power of graphs in various data association tasks. The supporting examples demonstrate the promise of Graph Data Analytics in modeling structural and functional/semantic inferences. At the same time, Part I serves as a basis for Part II and Part III which deal with theory, methods and applications of processing Data on Graphs and Graph Topology Learning from data. Contents 1 Introduction 2 2 Graph Definitions and Properties 3 2.1 Basic Definitions. .. .. .. .. .. .. .. 3 2.2 Some Frequently Used Graph Topologies. . 5 2.3 Properties of Graphs and Associated Matrices 7 3 Spectral Decomposition of Graph Matrices 10 3.

Spectral Estimation for Graph Signals Using Reed-Solomon Decoding

2018

Spectral estimation, coding theory and compressed sensing are three important sub-fields of signal processing and information theory. Although these fields developed fairly independently, several important connections between them have been identified. One notable connection between Reed-Solomon(RS) decoding, spectral estimation, and Prony's method of curve fitting was observed by Wolf in 1967. With the recent developments in the area of Graph Signal Processing(GSP), where the signals of interest have high dimensional and irregular structure, a natural and important question to consider is can these connections be extended to spectral estimation for graph signals? Recently, Marques et al, have shown that a bandlimited graph signal that is k-sparse in the Graph Fourier Transform (GFT) domain can be reconstructed from 2k measurements obtained using a dynamic sampling strategy. Inspired by this work, we establish a connection between coding theory and GSP to propose a sparse recove...

Incremental eigenpair computation for graph Laplacian matrices: theory and applications

Social Network Analysis and Mining, 2017

The smallest eigenvalues and the associated eigenvectors (i.e., eigenpairs) of a graph Laplacian matrix have been widely used in spectral clustering and community detection. However, in real-life applications the number of clusters or communities (say, K) is generally unknown a-priori. Consequently, the majority of the existing methods either choose K heuristically or they repeat the clustering method with different choices of K and accept the best clustering result. The first option, more often, yields suboptimal result, while the second option is computationally expensive. In this work, we propose an incremental method for constructing the eigenspectrum of the graph Laplacian matrix. This method leverages the eigenstructure of graph Laplacian matrix to obtain the K-th smallest eigenpair of the Laplacian matrix given a collection of all previously computed K − 1 smallest eigenpairs. Our proposed method adapts the Laplacian matrix such that the batch eigenvalue decomposition problem transforms into an efficient sequential leading eigenpair computation problem. As a practical application, we consider user-guided spectral clustering. Specifically, we demonstrate that users can utilize the proposed incremental method for effective eigenpair computation and for determining the desired number of clusters based on multiple clustering metrics.

Adaptive Graph Filters in Reproducing Kernel Hilbert Spaces: Design and Performance Analysis

IEEE Transactions on Signal and Information Processing over Networks, 2021

This paper develops adaptive graph filters that operate in reproducing kernel Hilbert spaces. We consider both centralized and fully distributed implementations. We first define nonlinear graph filters that operate on graph-shifted versions of the input signal. We then propose a centralized graph kernel least mean squares (GKLMS) algorithm to identify nonlinear graph filters' model parameters. To reduce the dictionary size of the centralized GKLMS, we apply the principles of coherence check and random Fourier features (RFF). The resulting algorithms have performance close to that of the GKLMS algorithm. Additionally, we leverage the graph structure to derive the distributed graph diffusion KLMS (GDKLMS) algorithms. We show that, unlike the coherence check-based approach, the GDKLMS based on RFF avoids the use of a pre-trained dictionary through its data-independent fixed structure. We conduct a detailed performance study of the proposed RFF-based GDKLMS, and the conditions for its convergence both in mean and mean-squared senses are derived. Extensive numerical simulations show that GKLMS and GDKLMS can successfully identify nonlinear graph filters and adapt to model changes. Furthermore, RFF-based strategies show faster convergence for model identification and exhibit better tracking performance in model-changing scenarios.

Spectral sparsification of graphs

Communications of the ACM, 2013

We introduce a new notion of graph sparsification based on spectral similarity of graph Laplacians: spectral sparsification requires that the Laplacian quadratic form of the sparsifier approximate that of the original. This is equivalent to saying that the Laplacian of the sparsifier is a good preconditioner for the Laplacian of the original.

Graph Learning Under Spectral Sparsity Constraints

ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2021

Graph inference plays an essential role in machine learning, pattern recognition, and classification. Signal processing based approaches in literature generally assume some variational property of the observed data on the graph. We make a case for inferring graphs on which the observed data has high variation. We propose a signal processing based inference model that allows for wideband frequency variation in the data and propose an algorithm for graph inference. The proposed inference algorithm consists of two steps: 1) learning orthogonal eigenvectors of a graph from the data; 2) recovering the adjacency matrix of the graph topology from the given graph eigenvectors. The first step is solved by an iterative algorithm with a closed-form solution. In the second step, the adjacency matrix is inferred from the eigenvectors by solving a convex optimization problem. Numerical results on synthetic data show the proposed inference algorithm can effectively capture the meaningful graph topology from observed data under the wideband assumption.