Graph Autoencoders with Deconvolutional Networks

1 INTRODUCTION . Autoencoders have demonstrated excellent performance on tasks such as unsupervised representation learning ( Bengio , 2009 ) and de-noising ( Vincent et al. , 2010 ) . Recently , several studies ( Zeiler & Fergus , 2014 ; Long et al. , 2015 ) have demonstrated that the performance of autoencoders can be further improved by encoding with Convolutional Networks and decoding with Deconvolutional Networks ( Zeiler et al. , 2010 ) . Notably , Noh et al . ( 2015 ) present a novel symmetric architecture that provides a bottom-up mapping from input signals to latent hierarchical feature space with { convolution , pooling } operations and then maps the latent representation back to the input space with { deconvolution , unpooling } operations . While this architecture has been successful when processing features with structures existed in the Euclidean space ( e.g. , images ) , recently there has been a surging interest in applying such a framework on non-Euclidean data like graphs . However , extending this autoencoder framework to graph-structured data requires Graph Deconvolutional operations , which remains open-ended and hasn ’ t been well-studied as opposed to the large body of works that have already been proposed for Graph Convolutional Networks ( Defferrard et al. , 2016 ; Kipf & Welling , 2017 ) . In this paper , we study the characteristics of Graph Deconvolutional Networks ( GDNs ) , and observe de-noising to be the key for effective deconvolutional operations . Therefore , we propose a wavelet-based module ( Hammond et al. , 2011 ) that serves as a de-noising mechanism after the signals reconstructed in the spectral domain ( Shuman et al. , 2013 ) for deconvolutional networks . Most GCNs proposed by prior arts , e.g. , Cheby-GCN ( Defferrard et al. , 2016 ) and GCN ( Kipf & Welling , 2017 ) , exploit spectral graph convolutions ( Shuman et al. , 2013 ) and Chebyshev polynomials ( Hammond et al. , 2011 ) to retain coarse-grained information and avoid explicit eigendecomposition of the graph Laplacian . Until recently , Wu et al . ( 2019 ) and Donnat et al . ( 2018 ) have noticed that GCN acts as a low pass filter in spectral domain and retains smoothed representations . Inspired by prior arts in the domain of signal deconvolution ( Kundur & Hatzinakos , 1996 ) , we propose to design a GDN by using high pass filters as the counterpart of low pass filters embodied in GCNs . Due to the nature of signal deconvolution being ill-posed , several prior arts ( Donoho & Johnstone , 1994 ; Figueiredo & Nowak , 2003 ) rely on transforming these signals into another domain ( e.g. , spectral domain ) where the problem can be better posed and resolved . Furthermore , Neelamani et al . ( 2004 ) observe inverse filters in spectral domain may amplify the noise , and we observe the same phenomenon for GDNs . Therefore , inspired by their proposed hybrid spectralwavelet method—inverse signal reconstruction in spectral domain followed by a de-noising step in wavelet domain—we introduce a spectral-wavelet GDN to decode the smoothed representations into the input graph signals . The proposed spectral-wavelet GDN employs spectral graph convolutions with a high pass filter to obtain inversed signals and then de-noises the inversed signals in wavelet domain . In addition , we apply Maclaurin series as a fast approximation technique to compute both high pass filters and wavelet kernels ( Donnat et al. , 2018 ) . With the proposed spectral-wavelet GDN , we further propose a graph autoencoder ( GAE ) framework that resembles the symmetric fashion of architectures ( Noh et al. , 2015 ) . We then evaluate the effectiveness of the proposed GAE framework with three popular and important tasks : unsupervised graph-level representation ( Sun et al. , 2020 ) , social recommendation ( Jamali & Ester , 2010 ) and graph generation . In the first task , the proposed GAE outperforms the state-of-the-arts on graph classification in an unsupervised fashion , along with a significant improvement on running time . In the second task , the performance of our proposed GAE is on par with the state-of-the-arts on the recommendation accuracy ; at the meantime , the proposed GAE demonstrates strong robustness against rating noises and achieves the best recommendation diversification ( Ziegler et al. , 2005 ) . In the third task , our proposed GDN can enhance the generation performance of popular variational autoencoder frameworks including VGAE ( Kipf & Welling , 2016 ) and Graphite ( Grover et al. , 2019 ) . 2 RELATED WORK . Deconvolutional networks The area of signal deconvolution ( Kundur & Hatzinakos , 1996 ) has a long history in the signal processing community and is about the process of estimating the true signals given the degraded or smoothed signal characteristics ( Banham & Katsaggelos , 1997 ) . Later deep learning studies ( Zeiler et al. , 2010 ; Noh et al. , 2015 ) consider deconvolutional networks as the opposite operation for Convolutional Neural Networks ( CNNs ) and have mainly focused on Euclidean structures , e.g. , image . Some work ( Dumoulin & Visin , 2016 ) notices Zeiler et al . ( 2010 ) is in essence a transposed convolution network as it differs from what is used in the signal processing community . For deconvolutional networks in non-Euclidean structures like graphs , the study is still sparse . Feizi et al . ( 2013 ) propose the network deconvolution as inferring the true network given partially observed structure . It relies on explicit eigen-decomposition and can not be used as the counterpart for GCN . Yang & Segarra ( 2018 ) formulate the deconvolution as a pre-processing step on the observed signals , in order to improve classification accuracy . Zhang et al . ( 2020 ) consider recovering graph signals from the latent representation . However , it just adopts the filter design used in GCN and sheds little insight into the internal operation of GDN . Graph autoencoders Since the introduction of Graph Neural Networks ( GNNs ) ( Kipf & Welling , 2017 ; Defferrard et al. , 2016 ) and autoencoders ( AEs ) , many studies ( Kipf & Welling , 2016 ; Grover et al. , 2019 ) have used GNNs and AEs to encode to and decode from latent representations . Recently graph pooling has emerged as a research topic that also contributes to the development of graph autoencoders . Common practices include DIFFPOOL ( Ying et al. , 2018 ) , SAGPool ( Lee et al. , 2019 ) , MinCut-Pool ( Bianchi et al. , 2020 ) . Although some encouraging progress has been achieved , there is still no work about graph deconvolution that can up-sample latent feature maps to restore their original resolutions ( Gao & Ji , 2019 ) . In this regard , current graph autoencoders bypass the difficulty via ( 1 ) non-parameterized decoders ( Kipf & Welling , 2016 ; Deng et al. , 2020 ; Li et al. , 2020 ) , ( 2 ) GCN decoders ( Grover et al. , 2019 ; Gao & Ji , 2019 ) , and ( 3 ) multilayer perceptron ( MLP ) decoders ( Simonovsky & Komodakis , 2018 ) . 3 GRAPH AUTOENCODER FRAMEWORK . Formally , we are given an undirected , unweighted graph G = ( V , A , X ) . V is the node set and N = |V | denotes the number of nodes . The adjacency matrix A ∈ RN×N represents the graph structure . The feature matrix X ∈ RN×d represents the node attributes . Our goal is to learn an encoder and a decoder to map between the space of graph G and their latent factors Gpool = ( V pool , Apool , Z ) . We show a schematic diagram of our proposed framework in Figure 1 . 3.1 ENCODER . Our encoder consists of several layers of Graph Convolutional Networks ( GCNs ) ( Kipf & Welling , 2017 ) and a pooling layer , to produce coarser representations of the input graphs . Convolution The convolutional layers are used to derive smoothed node representations , such that nodes that are similar in topological space should be close enough in Euclidean space . H = GCN ( A , X ) , ( 1 ) where H ∈ RN×v denotes smoothed node representations . Specifically , Wu et al . ( 2019 ) show that GCN is a low pass filter in spectral domain with gc ( λi ) = 1−λi , where { λi } Ni=1 are the eigenvalues of the normalized Laplacian matrix Lsym = D− 1 2LD− 1 2 , L and D are the Laplacian and degree matrices of the input graph A respectively . Pooling We follow Lee et al . ( 2019 ) and Li et al . ( 2019 ) by using the self attention mechanism to pool the fine-grained graph into coarse-grained representations , S = softmax ( tanh ( HW1 ) W2 ) , ( 2 ) where W1 ∈ Rv×d and W2 ∈ Rd×K are two weight matrices , W1 is used for feature transformation and W2 is used to infer the membership of each node with respect to each cluster Vk . Similar to Ying et al . ( 2018 ) , we compute the coarsed graph structure Apool ∈ RK×K and feature representation Z ∈ RK×v as follows : Z = S > H ; Apool = S > AS . ( 3 ) Note here ( Z , Apool ) is size invariant and permutation invariant , as pointed out by Li et al . ( 2019 ) . 3.2 DECODER . Our decoder consists of an unpooling layer and several layers of Graph Deconvolutional Networks ( GDNs ) , to produce fine-grained graphs from the encoded Gpool . Unpooling We follow Bianchi et al . ( 2020 ) to upscale the coarsened graph back to its original size , H ′ = SZ ; A′ = SApoolS > . ( 4 ) Deconvolution The deconvolutional layers are used to recover the original graph features given smoothed node representations , X ′ = GDN ( A′ , H ′ ) , ( 5 ) where X ′ ∈ RN×d denotes the recovered graph features . We shall further discuss our design of GDN in Section 4 . 3.3 THE LOSS FUNCTION . The overall reconstruction loss is a weighted sum of structure reconstruction loss and feature reconstruction loss . L = λAf ( A , A′ ) + λXf ( X , X ′ ) , ( 6 ) where f ( · , · ) denotes a differential distance metric , e.g. , f ( · , · ) = MSE ( · , · ) for continuous input , and MSE ( · , · ) represents mean squared error . 4 GRAPH DECONVOLUTIONAL NETWORKS . In this section , we present our design of Graph Deconvolutional Networks ( GDNs ) . A naive deconvolutional nets can be obtained using the inverse operator g−1c ( λi ) = 1 1−λi in spectral domain . Unfortunately , inverse operation results in a high pass filter and may amplify the noise ( Donoho & Johnstone , 1994 ; Figueiredo & Nowak , 2003 ) . In this regard , we propose an efficient , hybrid spectral-wavelet deconvolutional network that performs inverse signal recovery in spectral domain first , and then conducts a de-noising step in wavelet domain to remove the amplified noise ( Neelamani et al. , 2004 ) . 4.1 INVERSE OF GCN . In order to recover graph signals from the latent representation computed by GCN encoder , we proposed a naive approach–inverse GCN with the inverse filter as g−1c ( λi ) = 1 1−λi in spectral domain . The spectral graph convolution on a signal x ∈ RN is defined as : g−1c ∗ x = Udiag ( g−1c ( λ1 ) , . . . , g−1c ( λN ) ) U > x = Ug−1c ( Λ ) U > x , ( 7 ) where U is the eigenvector matrix of the normalized graph Laplacian Lsym = UΛU > . Then , we apply Maclaurin series approximation on g−1c ( Λ ) = ∑∞ n=0 Λ n and get a fast algorithm as below : g−1c ∗ x = U ∑∞ n=0 Λ nU > x = ∑∞ n=0 L n symx . ( 8 ) As in GCN ( Kipf & Welling , 2017 ) , when the first order approximation is used to address overfitting , we derive a spectral filter with g−1c ( λi ) = 1 + λi , which is apparently a high pass filter . Following GCN ( Kipf & Welling , 2017 ) , a feature transformation is applied to increase filter strength . Recap the GDN in Section 3.2 , the inverse version of GCN can be written as : M = ( IN + L ′ sym ) H ′W3 , ( 9 ) where L′sym is the corresponding normalized graph Laplacian matrix for A ′ , H ′ is the smoothed representations and W3 is the parameter set to be learned . Compared with directly using GCN for signal reconstruction in Zhang et al . ( 2020 ) , the proposed inverse GCN demonstrates its efficacy in recovering the high frequency signals of the graph , as shown in Figure 2 ( b ) and ( d ) . We shall further discuss this point in Section 4.3 .

Summary: The main contribution of this paper is that the authors design a graph deconvolutional network that combines inverse filters in the spectral domain and de-noising layers in the wavelet domain. Further graph autoencoders are proposed based on the graph convolutional networks and the graph deconvolutional networks. Many experiments are done and many previous methods are compared to show the effectiveness of the proposed networks.