ABSTRACT
Graph kernels are widely used for measuring the similarity between graphs. Many existing graph kernels, which focus on local patterns within graphs rather than their global properties, suffer from significant structure information loss when representing graphs. Some recent global graph kernels, which utilizes the alignment of geometric node embeddings of graphs, yield state-of-the-art performance. However, these graph kernels are not necessarily positive-definite. More importantly, computing the graph kernel matrix will have at least quadratic time complexity in terms of the number and the size of the graphs. In this paper, we propose a new family of global alignment graph kernels, which take into account the global properties of graphs by using geometric node embeddings and an associated node transportation based on earth mover's distance. Compared to existing global kernels, the proposed kernel is positive-definite. Our graph kernel is obtained by defining a distribution over random graphs, which can naturally yield random feature approximations. The random feature approximations lead to our graph embeddings, which is named as "random graph embeddings" (RGE). In particular, RGE is shown to achieve (quasi-)linear scalability with respect to the number and the size of the graphs. The experimental results on nine benchmark datasets demonstrate that RGE outperforms or matches twelve state-of-the-art graph classification algorithms.
- Rami Al-Rfou, Dustin Zelle, and Bryan Perozzi. 2019. DDGK: Learning Graph Representations for Deep Divergence Graph Kernels. arXiv:1904.09671 (2019). Google ScholarDigital Library
- James Atwood, Siddharth Pal, Don Towsley, and Ananthram Swami. 2016. Sparse Diffusion-Convolutional Neural Networks. In NIPS. Google ScholarDigital Library
- Francis Bach. 2017. On the equivalence between kernel quadrature rules and random feature expansions. Journal of Machine Learning Research 18, 21 (2017), 1--38. Google ScholarDigital Library
- Karsten M Borgwardt and Hans-Peter Kriegel. 2005. Shortest-path kernels on graphs. In Data Mining, Fifth IEEE International Conference on. IEEE, 8--pp. Google ScholarDigital Library
- Francois Bourgeois and Jean-Claude Lassalle. 1971. An extension of the Munkres algorithm for the assignment problem to rectangular matrices. Commun. ACM 14, 12 (1971), 802--804. Google ScholarDigital Library
- Pin-Yu Chen and Lingfei Wu. 2017. Revisiting spectral graph clustering with generative community models. In ICDM. 51--60.Google Scholar
- Rong-En Fan, Kai-Wei Chang, Cho-Jui Hsieh, Xiang-Rui Wang, and Chih-Jen Lin. 2008. LIBLINEAR: A library for large linear classification. Journal of machine learning research 9, Aug (2008), 1871--1874. Google ScholarDigital Library
- Matthew Fisher, Manolis Savva, and Pat Hanrahan. 2011. Characterizing struc- tural relationships in scenes using graph kernels. ACM Transactions on Graphics (TOG) 30, 4 (2011), 34. Google ScholarDigital Library
- Thomas Gartner, Peter Flach, and Stefan Wrobel. 2003. On graph kernels: Hard- ness results and efficient alternatives. In Learning Theory and Kernel Machines. Springer, 129--143.Google Scholar
- David Haussler. 1999. Convolution kernels on discrete structures. Technical Report. Department of Computer Science, University of California at Santa Cruz.Google Scholar
- Frank L Hitchcock. 1941. The distribution of a product from several sources to numerous localities. Studies in Applied Mathematics 20, 1--4 (1941), 224--230.Google Scholar
- Tama's Horvath, Thomas Gartner, and Stefan Wrobel. 2004. Cyclic pattern kernels for predictive graph mining. In KDD. ACM, 158--167. Google ScholarDigital Library
- Catalin Ionescu, Alin Popa, and Cristian Sminchisescu. 2017. Large-scale data- dependent kernel approximation. In Artificial Intelligence and Statistics. 19--27.Google Scholar
- Fredrik Johansson, Vinay Jethava, Devdatt Dubhashi, and Chiranjib Bhat- tacharyya. 2014. Global graph kernels using geometric embeddings. In ICML. Google ScholarDigital Library
- Fredrik D Johansson and Devdatt Dubhashi. 2015. Learning with similarity functions on graphs using matchings of geometric embeddings. In KDD. ACM, 467--476. Google ScholarDigital Library
- Risi Kondor and Horace Pan. 2016. The multiscale laplacian graph kernel. In NIPS. 2990--2998. Google ScholarDigital Library
- Nils M Kriege, Pierre-Louis Giscard, and Richard Wilson. 2016. On valid optimal assignment kernels and applications to graph classification. In NIPS. 1623--1631. {18} Matt Kusner, Yu Sun, Nicholas Kolkin, and Kilian Weinberger. 2015. From word embeddings to document distances. In ICML. 957--966. Google ScholarDigital Library
- Quoc Le, Tama's Sarlo's, and Alex Smola. 2013. Fastfood-approximating kernel expansions in loglinear time. In ICML, Vol. 85. Google ScholarDigital Library
- La'szlo Lova'sz. 1979. On the Shannon capacity of a graph. IEEE Transactions on Information theory 25, 1 (1979), 1--7. Google ScholarDigital Library
- Fatemeh Mokhtari and Gholam-Ali Hossein-Zadeh. 2013. Decoding brain states using backward edge elimination and graph kernels in fMRI connectivity net- works. Journal of neuroscience methods 212, 2 (2013), 259--268.Google ScholarCross Ref
- Mathias Niepert, Mohamed Ahmed, and Konstantin Kutzkov. 2016. Learning convolutional neural networks for graphs. In ICML. 2014--2023. Google ScholarDigital Library
- Giannis Nikolentzos, Polykarpos Meladianos, and Michalis Vazirgiannis. 2017. Matching Node Embeddings for Graph Similarity.. In AAAI. 2429--2435. Google ScholarDigital Library
- Ali Rahimi and Benjamin Recht. 2008. Random features for large-scale kernel machines. In NIPS. 1177--1184. Google ScholarDigital Library
- Liva Ralaivola, Sanjay J Swamidass, Hiroto Saigo, and Pierre Baldi. 2005. Graph kernels for chemical informatics. Neural networks 18, 8 (2005), 1093--1110. Google ScholarDigital Library
- Yossi Rubner, Carlo Tomasi, and Leonidas J Guibas. 2000. The earth mover's distance as a metric for image retrieval. International journal of computer vision 40, 2 (2000), 99--121. Google ScholarDigital Library
- Alessandro Rudi and Lorenzo Rosasco. 2017. Generalization properties of learning with random features. In NIPS. 3218--3228. Google ScholarDigital Library
- Nino Shervashidze and Karsten M Borgwardt. 2009. Fast subtree kernels on graphs. In NIPS. 1660--1668. Google ScholarDigital Library
- Nino Shervashidze, Pascal Schweitzer, Erik Jan van Leeuwen, Kurt Mehlhorn, and Karsten M Borgwardt. 2011. Weisfeiler-lehman graph kernels. Journal of Machine Learning Research 12, Sep (2011), 2539--2561. Google ScholarDigital Library
- NinoShervashidze,SVNVishwanathan,TobiasPetri,KurtMehlhorn,andKarsten Borgwardt. 2009. Efficient graphlet kernels for large graph comparison. In AIStats. 488--495.Google Scholar
- Aman Sinha and John C Duchi. 2016. Learning kernels with random features. In NIPS. 1298--1306. Google ScholarDigital Library
- Justin Solomon, Raif Rustamov, Leonidas Guibas, and Adrian Butscher. 2016. Continuous-flow graph transportation distances. arXiv:1603.06927 (2016).Google Scholar
- Andreas Stathopoulos and James R McCombs. 2010. PRIMME: preconditioned iterative multimethod eigensolveraATmethods and software description. ACM Transactions on Mathematical Software (TOMS) 37, 2 (2010), 21. Google ScholarDigital Library
- S Vichy N Vishwanathan, Nicol N Schraudolph, Risi Kondor, and Karsten M Borgwardt. 2010. Graph kernels. Journal of Machine Learning Research 11 (2010), 1201--1242. Google ScholarCross Ref
- Ulrike Von Luxburg. 2007. A tutorial on spectral clustering. Statistics and computing 17, 4 (2007), 395--416. Google ScholarDigital Library
- Cynthia Wagner, Gerard Wagener, Radu State, and Thomas Engel. 2009. Malware analysis with graph kernels and support vector machines. In Malicious and Unwanted Software, 2009 4th International Conference on. IEEE, 63--68.Google ScholarCross Ref
- Ling Wang and Hichem Sahbi. 2013. Directed acyclic graph kernels for action recognition. In ICCV. IEEE, 3168--3175. Google ScholarDigital Library
- Bo Wu, Yang Liu, Bo Lang, and Lei Huang. 2017. DGCNN: Disordered Graph Convolutional Neural Network Based on the Gaussian Mixture Model. arXiv:1712.03563 (2017).Google Scholar
- Lingfei Wu, Pin-Yu Chen, Ian En-Hsu Yen, Fangli Xu, Yinglong Xia, and Charu Aggarwal. 2018. Scalable spectral clustering using random binning features. In KDD. ACM, 2506--2515. Google ScholarDigital Library
- Lingfei Wu, Eloy Romero, and Andreas Stathopoulos. 2017. PRIMME_SVDS: A high-performance preconditioned SVD solver for accurate large-scale computa- tions. SIAM Journal on Scientific Computing 39, 5 (2017), S248--S271.Google ScholarCross Ref
- Lingfei Wu, Ian EH Yen, Jie Chen, and Rui Yan. 2016. Revisiting random binning features: Fast convergence and strong parallelizability. In KDD. ACM, 1265--1274. Google ScholarDigital Library
- Lingfei Wu, Ian En-Hsu Yen, Fangli Xu, Pradeep Ravikuma, and Michael Wit- brock. 2018. D2KE: From Distance to Kernel and Embedding. arXiv preprint arXiv:1802.04956 (2018).Google Scholar
- Lingfei Wu, Ian En-Hsu Yen, Jinfeng Yi, Fangli Xu, Qi Lei, and Michael Witbrock. 2018. Random Warping Series: A Random Features Method for Time-Series Embedding. In International Conference on Artificial Intelligence and Statistics. 793--802.Google Scholar
- Pinar Yanardag and SVN Vishwanathan. 2015. Deep graph kernels. In KDD. ACM, 1365--1374. Google ScholarDigital Library
- Pinar Yanardag and SVN Vishwanathan. 2015. A structural smoothing framework for robust graph comparison. In NIPS. 2134--2142. Google ScholarDigital Library
- Zhen Zhang, Mianzhi Wang, Yijian Xiang, Yan Huang, and Arye Nehorai. 2018. RetGK: Graph Kernels based on Return Probabilities of Random Walks. In NIPS. 3968--3978. Google ScholarDigital Library
Index Terms
- Scalable Global Alignment Graph Kernel Using Random Features: From Node Embedding to Graph Embedding
Recommendations
A generalized Weisfeiler-Lehman graph kernel
AbstractAfter more than one decade, Weisfeiler-Lehman graph kernels are still among the most prevalent graph kernels due to their remarkable predictive performance and time complexity. They are based on a fast iterative partitioning of vertices, ...
Investigation of kernel functions in EDA-GK
GECCO '18: Proceedings of the Genetic and Evolutionary Computation Conference CompanionWe have proposed EDA-GK, Estimation of Distribution Algorithms with Graph Kernels. The EDA-GK is designed for solving graph-related problems, where individuals can be represented by graphs. By using graph kernels, the EDA-GK can be solved for graph-...
Dual-decoder graph autoencoder for unsupervised graph representation learning
AbstractUnsupervised graph representation learning is a challenging task that embeds graph data into a low-dimensional space without label guidance. Recently, graph autoencoders have been proven to be an effective way to solve this problem in ...
Comments