Quantization in Graph Convolutional Neural Networks Leila Ben Saad and Baltasar Beferull-Lozano WISENET Center, Department of ICT, University of Agder, Grimstad, Norway Abstract—By replacing classical convolutions with graph fil- ters, graph convolutional neural networks (GNNs) have emerged as powerful tools to learn a nonlinear mapping for data defined over graphs and address a variety of tasks encountered in many applications. GNNs inherit the distributed implementation of graph filters, where local exchanges among neighbor nodes are performed. In such distributed setting, the quantization can play a fundamental role to save communication and energy resources prior to data transmission, in scenarios where nodes are resource constrained. In this paper, we propose a quantized GNN architecture based on distributed graph filters for signals defined on graphs and analyze how the quantization noise can affect its performance. We show also that the expected error due to quantization at the GNN output is upper-bounded and the use of a decreasing quantization stepsize leads to more accuracy. The performance of the proposed schemes is evaluated by numerical experiments for the application of source localization. Index Terms—Graph neural networks; Graph signal process- ing; Graph filters; Quantization. I. I NTRODUCTION Recently, graph convolutional neural networks (GNNs) [1– 3] have emerged as a way to generalize and extend the convolutional neural networks (CNNs) to data supported on graphs by processing signals defined in irregular domains and replacing classical convolutions with graph filters [4–6]. GNNs offer the possibility to learn a nonlinear mapping for data defined over graphs that can be encountered in many applications, such as social networks, sensor networks and recommendation systems. GNNs are formed by layers of graph filters followed by a pointwise nonlinearity. By using Finite Impulse Response (FIR) graph filters [4–6], GNNs inherit their distributed implementation, which is very important to ensure the scalability and robustness to possible node failures. FIR graph filters can have different forms of implementation [5, 6]: node-invariant, node-variant and edge-variant. In such graph filters, each node can communicate its input signal through local exchanges with neighbors in a finite number of iterations. However, when implemented over distributed networks with constrained node resources, the graph filters in GNNs have to deal with the limited energy, processing, and communication capabilities. The quantization is recognized as an effective approach to save communication and energy resources prior to data transmission. Although the quantization has been well studied in neural networks [7, 8] and CNNs [9–11], very few works [12, 13] have explored the quantization problem This work was supported by the PETROMAKS Smart-Rig grant 244205/E30, the TOPPFORSK WISECART grant 250910/F20 and the IK- TPLUSS INDURB grant 270730/O70 from the Research Council of Norway. © 2021 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. DOI: 10.23919/EUSIPCO54536.2021.9615973 in graph neural networks. In [12], a method for training a quantized GNN is proposed, while in [13] a GNN quantization algorithm and a fine-tuning scheme are proposed to solve the GNN memory problem. To the best of our knowledge, there is no work investigating the quantization for GNNs via distributed graph filters for signals supported on graphs. In this paper, we propose a quantized GNN architecture built on distributed graph filters and analyze how the quantization errors accumulated over the different layers can affect its final output. Considering the different forms of FIR graph filters implementation, we show how the expected error due to quantization at the GNN output is upper-bounded and the use of a decreasing quantization stepsize leads to more accu- racy as compared to a fixed quantization stepsize. Numerical experiments have been conducted to corroborate our findings. The remainder of this paper is organized as follows. Section II presents the background material. Section III introduces the quantization in GNNs for signals supported on graphs and Section IV analyzes its impacts. Section V presents the numerical results. Section VI concludes the paper. Notation: Vectors (respectively matrices) are denoted by bold lowercase (uppercase) letters. We denote by ∥v∥ the l 2 - norm of vector v whereas by ∥M∥ the spectral norm of matrix M. We indicate by tr(·), diag(·) and Σ x the trace operator, the diagonal matrix and the covariance matrix of a random process x, respectively. II. BACKGROUND Consider an undirected graph G = (V , E ) where V = {1,...,N } is the set of N vertices and E is the set of edges such that if there is a link from node j to node i, then (j, i) ∈E . The structure of G is generally represented by the graph-shift operator S, which is an N × N matrix such that its entries are non-zero only if i = j or if (j, i) ∈E . Common choices of the graph-shift operator are the graph adjacency matrix A, the graph Laplacian L or their normalized versions. We define on the vertices of G a graph signal as a mapping x : V→ R, which can be represented as a vector x =[x 1 , ..., x N ] ⊤ ∈ R N , where the i-th entry represents the signal value at node i. FIR graph filters. A linear graph filter (GF) [4] H(S): R N → R N is a function of the shift operator S, and where a graph signal x is taken as an input and another graph signal y = H(S)x is produced as an output. One common form for H(S) is the so-called node-invariant GF, which is a polynomial in S with output: y = H(S)x = K X k=0 h k S k x (1) where h 0 ,...,h K are the scalar filter coefficients. To approximate a broader class of operations, the node- variant GF [6] has been proposed and has as output: