Abstract
Graph Convolutional Networks (GCNs) have attracted broad attention from industry and academia, for their excellent expressive power in terms of modeling the irregular data, e.g., skeletal data and graph-structured data. The most effective existing model may be the fully hyperbolic graph neural network. However, it involves a large number of parameters, thus consuming considerable computing resources. In this paper, we propose a model based on adaptive frequency filter and corresponding optimizer in hyperbolic space. The adaptive frequency can learn the different frequency components of the embeddings of the nodes in graph, which adaptively adjust the beneficial signals of high-frequency and low-frequency. And the optimizer is based on a subset of the orthogonal constraint, which is dedicated for the adaptive frequency with less parameters. Moreover, our model bridges the gap of hyperbolic space and the spectral space for exploring the underlying semantics of the node and relation embeddings of graph. Consequently, our model needs only to optimize the less parameters in hyperbolic space and meanwhile prevent the distortion caused by conventional manifold GCN. Experimental results show that our method achieves substantial improvements and outperforms the state-of-the-art performance in terms of node classification.
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The datasets generated and analyzed during the current study are available from the corresponding author on reasonable request.
References
Wang H, Xu T, Liu Q, Lian D, Chen E, Du D, Wu H, Su W (2019) MCNE: an end-to-end framework for learning multiple conditional network representations of social network. In: Proceedings of the 25th ACM SIGKDD international conference on knowledge discovery & data mining, pp 1064–1072
Ali Z, Qi G, Muhammad K, Bhattacharyya S, Ullah I, Abro W (2022) Citation recommendation employing heterogeneous bibliographic network embedding. Neural Comput Appl 34(13):10229–10242
Zhang X-M, Liang L, Liu L, Tang M-J (2021) Graph neural networks and their current applications in bioinformatics. Front Genetics 12:690049
Rahevar M, Ganatra A (2023) Spatial-temporal gated graph attention network for skeleton-based action recognition. Pattern Anal Appl 26:1–11
Hamilton WL, Ying R, Leskovec J (2017) Inductive representation learning on large graphs. In: Proceedings of the 31st international conference on neural information processing systems, pp 1025–1035
Kipf TN, Welling M (2017) Semi-supervised classification with graph convolutional networks. In: Proceedings of the international conference on learning
Chen W, Fang W, Hu G, Mahoney MW (2013) On the hyperbolicity of small-world and treelike random graphs. Internet Math 9(4):434–491
Sarkar R (2011) Low distortion delaunay embedding of trees in hyperbolic plane. In: International symposium on graph drawing. Springer, pp. 355–366
Chami I, Ying Z, Ré C, Leskovec J (2019) Hyperbolic graph convolutional neural networks. Adv Neural Inf Process Syst 32:4868–4879
Nickel M, Kiela D (2017) Poincaré embeddings for learning hierarchical representations. Adv Neural Inf Process Syst 30:6338–6347
Krioukov D, Papadopoulos F, Kitsak M, Vahdat A, Boguná M (2010) Hyperbolic geometry of complex networks. Phys Rev E 82(3):036106
Muscoloni A, Thomas JM, Ciucci S, Bianconi G, Cannistraci CV (2017) Machine learning meets complex networks via coalescent embedding in the hyperbolic space. Nat Commun 8(1):1–19
Dai J, Wu Y, Gao Z, Jia Y (2021) A hyperbolic-to-hyperbolic graph convolutional network. In: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp 154–163
Dong Y, Ding K, Jalaian B, Ji S, Li J (2021) AdaGNN: graph neural networks with adaptive frequency response filter. In: Proceedings of the 30th ACM international conference on information & knowledge management, pp 392–401
Wu F, Souza A, Zhang T, Fifty C, Yu T, Weinberger K (2019) Simplifying graph convolutional networks. In: International conference on machine learning. PMLR, pp 6861–6871
Gori M, Monfardini G, Scarselli F (2005) A new model for learning in graph domains. In: Proceedings. 2005 IEEE international joint conference on neural networks, vol 2, pp 729–734
Scarselli F, Gori M, Tsoi AC, Hagenbuchner M, Monfardini G (2008) The graph neural network model. IEEE Trans Neural Netw 20(1):61–80
Bruna J, Zaremba W, Szlam A, LeCun Y (2014) Spectral networks and locally connected networks on graphs. In: Proceedings of the international conference on learning representations
Defferrard M, Bresson X, Vandergheynst P (2016) Convolutional neural networks on graphs with fast localized spectral filtering. Adv Neural Inf Process Syst 29
Duvenaud DK, Maclaurin D, Iparraguirre J, Bombarell R, Hirzel T, Aspuru-Guzik A, Adams RP (2015) Convolutional networks on graphs for learning molecular fingerprints. Adv Neural Inf Process Syst 28
Gilmer J, Schoenholz SS, Riley PF, Vinyals O, Dahl GE (2017) Neural message passing for quantum chemistry. In: International conference on machine learning. PMLR, pp. 1263–1272
Papadopoulos F, Kitsak M, Serrano M, Boguná M, Krioukov D (2012) Popularity versus similarity in growing networks. Nature 489(7417):537–540
Nickel M, Kiela D (2018) Learning continuous hierarchies in the Lorentz model of hyperbolic geometry. In: International conference on machine learning. PMLR, pp 3779–3788
Balazevic I, Allen C, Hospedales T (2019) Multi-relational Poincaré graph embeddings. Adv Neural Inf Process Syst 32
Sun Z, Chen M, Hu W, Wang C, Dai J, Zhang W (2020) Knowledge association with hyperbolic knowledge graph embeddings. In: EMNLP, pp 5704–5716
Khrulkov V, Mirvakhabova L, Ustinova E, Oseledets I, Lempitsky V (2020) Hyperbolic image embeddings. In: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp 6418–6428
Liu S, Chen J, Pan L, Ngo C-W, Chua T-S, Jiang Y-G (2020) Hyperbolic visual embedding learning for zero-shot recognition. In: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp 9273–9281
Bronstein MM, Bruna J, LeCun Y, Szlam A, Vandergheynst P (2017) Geometric deep learning: going beyond Euclidean data. IEEE Signal Process Mag 34(4):18–42
Liu Q, Nickel M, Kiela D (2019) Hyperbolic graph neural networks. Adv Neural Inf Process Syst 32
Zhuang C, Ma Q (2018) Dual graph convolutional networks for graph-based semi-supervised classification. In: Proceedings of the 2018 world wide web conference, pp 499–508
Atwood J, Towsley D (2016) Diffusion-convolutional neural networks. Adv Neural Inf Process Syst 29
Dong Y, Liu N, Jalaian B, Li J (2022) Edits: modeling and mitigating data bias for graph neural networks. In: Proceedings of the ACM web conference 2022, pp 1259–1269
Levie R, Monti F, Bresson X, Bronstein MM (2018) CayleyNets: graph convolutional neural networks with complex rational spectral filters. IEEE Trans Signal Process 67(1):97–109
Hoang N, Maehara T, Murata T (2021) Revisiting graph neural networks: graph filtering perspective. In: 2020 25th international conference on pattern recognition (ICPR). IEEE, pp 8376–8383
Bo D, Wang X, Shi C, Shen H (2021) Beyond low-frequency information in graph convolutional networks. arXiv preprint arXiv:2101.00797
Chen Y, Fan H, Xu B, Yan Z, Kalantidis Y, Rohrbach M, Yan S, Feng J (2019) Drop an octave: reducing spatial redundancy in convolutional neural networks with octave convolution. In: Proceedings of the IEEE/CVF international conference on computer vision, pp 3435–3444
Robbin JW, Salamon DA (2011) Introduction to differential geometry. ETH, Lecture Notes, preliminary version, 18
He K, Zhang X, Ren S, Sun J (2016) Deep residual learning for image recognition. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 770–778
Xu K, Hu W, Leskovec J, Jegelka S (2018) How powerful are graph neural networks? In: Proceedings of the 7th international conference on learning representations
Veličković P, Cucurull G, Casanova A, Romero A, Lio P, Bengio Y (2018) Graph attention networks. In: International conference on learning representations
Boothby WM, Boothby WM (2003) An introduction to differentiable manifolds and Riemannian geometry, revised. Gulf Professional Publishing 120
Fréchet M (1948) Les éléments aléatoires de nature quelconque dans un espace distancié. In: Annales de L’institut Henri Poincaré, vol 10, pp 215–310
Ungar AA (2005) Analytic hyperbolic geometry: mathematical foundations and applications. World Scientific
Shimizu R, Mukuta Y, Harada T (2021) Hyperbolic neural networks++. In: Proceedings of the international conference on learning representations
Anderson RM, May RM (1992) Infectious diseases of humans: dynamics and control. Oxford University Press
Zeng Z, Peng Q, Mou X, Wang Y, Li R (2023) Graph neural networks with high-order polynomial spectral filters. IEEE Trans Neural Netw Learn Syst. https://doi.org/10.1109/TNNLS.2023.3263676
Sen P, Namata G, Bilgic M, Getoor L, Galligher B, Eliassi-Rad T (2008) Collective classification in network data. AI Mag 29(3):93–93
Paszke A, Gross S, Chintala S, Chanan G, Yang E, DeVito Z, Lin Z, Desmaison A, Antiga L, Lerer A (2017) Automatic differentiation in PyTorch
Kingma DP, Ba J (2015) Adam: a method for stochastic optimization. In: Proceedings of ICLR
Van der Maaten L, Hinton G (2008) Visualizing data using t-SNE. J Mach Learn Res 9(11):2579
Singh R, Gill SS (2023) Edge AI: a survey. Internet of Things and Cyber-Physical Systems
Elias VRM, Gogineni VC, Martins WA, Werner S (2022) Kernel regression over graphs using random Fourier features. IEEE Trans Signal Process 70:936–949
Nikhitha NK, Afzal A, Asharaf S (2021) Deep kernel machines: a survey. Pattern Anal Appl 24:537–556
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This paper is supported by National Natural Science Foundation of China (Grant No. 62076193).
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Wei, F., Ping, M. & Mei, K. Adaptive frequency-based fully hyperbolic graph neural networks. Pattern Anal Applic 26, 1741–1751 (2023). https://doi.org/10.1007/s10044-023-01201-8
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DOI: https://doi.org/10.1007/s10044-023-01201-8