GRAPH SIGNAL PROCESSING: FOUNDATIONS AND EMERGING DIRECTIONS Yuichi Tanaka, Yonina C. Eldar, Antonio Ortega, and Gene Cheung Sampling Signals on Graphs From theory to applications he study of sampling signals on graphs, with the goal of building an analog of sampling for standard signals in the T time and spatial domains, has attracted considerable atten- tion recently. Beyond adding to the growing theory on graph signal processing (GSP), sampling on graphs has various promising applications. In this article, we review the current progress on sampling over graphs, focusing on theory and potential applications. Although most methodologies used in graph signal sam- pling are designed to parallel those used in sampling for standard signals, sampling theory for graph signals signifi- cantly differs from the theory of Shannon–Nyquist and shift- invariant (SI) sampling. This is due, in part, to the fact that the definitions of several important properties, such as shift invariance and bandlimitedness, are different in GSP systems. Throughout this review, we discuss similarities and differenc- es between standard and graph signal sampling and highlight open problems and challenges. Introduction Sampling is one of the fundamental tenets of digital signal pro- cessing (see [1] and the references therein). As such, it has been studied extensively for decades and continues to draw consid- erable research efforts. Standard sampling theory relies on concepts of frequency domain analysis, SI signals, and band- ©ISTOCKPHOTO.COM/ALISEFOX limitedness [1]. The sampling of time and spatial domain sig- nals in SI spaces is one of the most important building blocks of digital signal processing systems. However, in the big data era, the signals we need to process often have other types of connections and structure, such as network signals described by graphs. This article provides a comprehensive overview of the theo- ry and algorithms for the sampling of signals defined on graph domains, i.e., graph signals. GSP [2]–[4]—a fast developing field in the signal processing community—generalizes key sig- nal processing ideas for signals defined on regular domains to discrete-time signals defined over irregular domains described Digital Object Identifier 10.1109/MSP.2020.3016908 Date of current version: 28 October 2020 abstractly by graphs. GSP has found numerous promising 14 IEEE SIGNAL PROCESSING MAGAZINE | November 2020 | 1053-5888/20©2020IEEE Authorized licensed use limited to: Weizmann Institute of Science. Downloaded on November 01,2020 at 09:14:01 UTC from IEEE Xplore. Restrictions apply. applications across many engineering disciplines, including []X ij . A subvector of x is denoted xS with indicator index set S. NM# ||||RC# image processing, wireless communications, machine learn- Similarly, a submatrix of X ! R is denoted XRC ! R , ing, and data mining [2], [3], [5]. where indicator indices of its rows and columns are given by R Network data are pervasive and found in applications such and C, respectively; XRR is simply written as XR . as sensor, neuronal, transportation, and social networks. The number of nodes in such networks is often very large: process- Review: GSP and sampling in Hilbert spaces ing and storing all of the data as is can require huge computa- tion and storage resources, which may not be tolerable even The basics of GSP in modern, high-performance communication and computer We denote by GV= (,E) a graph where V and E are the sets systems. Therefore, it is often of interest to reduce the amount of vertices and edges, respectively. The number of vertices is of data while keeping the important information as much as N = | V | unless otherwise specified. We define an adjacency possible. Sampling of graph signals addresses this issue: how matrix W where entry []W mn represents the weight of the edge one can reduce the number of samples on a between vertices m and n; []W mn = 0 for graph and reconstruct the underlying signal, GSP has found numerous unconnected vertices. The degree matrix generalizing the standard sampling paradigm promising applications D is diagonal, with mth diagonal element to graph signals. across many engineering []DWmm = Rnm[ ] n . In this article, we con- Generalization of the sampling prob- disciplines, including sider undirected graphs without self-loops, lem to GSP raises a number of challenges. i.e., []WWmn = []nm and []W nn = 0 for all First, for a given graph and graph operator, image processing, m and n, but most theory and methods dis- the notion of frequency for graph signals wireless communications, cussed can be extended to signals on direct- is mathematically straightforward, but the machine learning, and ed graphs. connection of these frequencies to the actual data mining. GSP uses different variation operators [2], properties of signals of interest (and, thus, [3] depending on the application and assumed the practical meaning of concepts, such as bandlimitedness and signal and/or network models. Here, for concreteness, we focus on smoothness) is still being investigated. Second, periodic sam- the graph Laplacian LD:=-W or its symmetrically normal- pling, widely used in traditional signal processing, is not appli- ized version LD:.= --12//LD 12 The extension to other variation cable in the graph domain (e.g., it is unclear how to select “every operators (e.g., adjacency matrix) is possible with a proper modifi- other” sample). Although the theory of sampling on graphs and cation of the basic operations discussed in this section. Since L is a manifolds has been studied in the past (see [6], [7], and follow- real symmetric matrix, it always possesses an eigendecomposition < up works), early works have not considered the problems inher- LU= KU , where Uu= [,1 f,]uN is an orthonormal matrix ent in applications, e.g., how to select the best set of nodes from containing the eigenvectors ui, and K = diag(,mm1 f,)N con- a given graph. Therefore, developing practical techniques for sists of the eigenvalues mi. We refer to mi as the graph frequency. sampling set selection that can adapt to local graph topology is A graph signal x :V " R is a function that assigns a value to very important. Third, the work to date has mostly focused on each node. Graph signals can be written as vectors, x, in which the direct nodewise sampling, while there has been only limited nth element, xn[], represents the signal value at node n. Note that work on developing more advanced forms of sampling, e.g., any vertex labeling can be used, since a change in labeling simply adapting SI sampling [1] to the graph setting [8]. Finally, graph results in row/column permutation of the various matrices, their signal sampling and reconstruction algorithms must be imple- corresponding eigenvectors, and the vectors representing graph mented efficiently to achieve a good tradeoff between accuracy signals. The graph Fourier transform (GFT) is defined as and complexity. To address these challenges, various graph sampling N- 1 xit []==GHuxii,[/ u nxn][]. (1) approaches have recently been developed (e.g., [7] and [9]–[14]) n = 0 based on different notions of graph frequency, bandlimitedness, and shift invariance. For example, a common approach to define Other GFT definitions can also be used without changing the the graph frequency is based on the spectral decomposition of framework. In this article, for simplicity, we assume real-val- different variation operators, such as the adjacency matrix or ued signals. Although the GFT basis is real valued for undi- variants of graph Laplacians. The proposed reconstruction pro- rected graphs, extensions to complex-valued GSP systems are cedures in the literature differ in their objective functions, lead- straightforward. ing to a tradeoff between accuracy and complexity. Our goal is A linear graph filter is defined by G ! RNN# , which, applied to provide a broad overview of existing techniques, highlighting to x, produces an output what is known to date to inspire further research on sampling yG= x. (2) over graphs and its use in a broad class of applications in signal processing and machine learning. Vertex- and frequency-domain graph filter designs considered In what follows, we use boldfaced lowercase (uppercase) sym- in the literature both lead to filters G that depend on the struc- bols to represent vectors (matrices); the ith element in a vector x ture of the graph G. Vertex-domain filters are defined as poly- is xi[] or xi; and the ith row, jth column of a matrix X is given by nomials of the variation operator, i.e., IEEE SIGNAL PROCESSING MAGAZINE | November 2020 | 15 Authorized licensed use limited to: Weizmann Institute of Science. Downloaded on November 01,2020 at 09:14:01 UTC from IEEE Xplore. Restrictions apply. P p is the sampled cross correlation. Thus, we may view sampling yG==xLc p x, (3) / in the Fourier domain as multiplying the input spectrum by the eop = 0 filter’s frequency response and subsequently aliasing the result where the output at each vertex is a linear combination of sig- with uniform intervals that depend on the sampling period. In nal values in its P-hop neighborhood. In frequency-domain fil- BL sampling, st()-=sinc(/tT), where sinc()tt= sin()rr/( t), ter design, G is chosen to be diagonalized by U so that while s(t) may be chosen more generally in the generalized sam- pling framework. yG==xUgt ()K Ux< , (4) Recovery of the signal from its samples c is represented as * where ggtt()K :(= diag ()mm1 ,,f gt ()N ) is the graph frequen- xWu ==Hc WH()Sx, (8) cy response. Filtering via (4) is analogous to filtering in the Fourier domain for conventional signals. When there where W is a set transformation corresponding to a basis exist repeated eigenvalues mmij= , their graph frequency {}wn for the reconstruction space, which spans a closed responses must be the same, i.e., ggtt()mmij= ().