ハイパーグラフ上の熱とそのネットワーク解析への応用
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| Published in | 応用数理 Vol. 31; no. 2; pp. 2 - 10 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | Japanese |
| Published |
一般社団法人 日本応用数理学会
24.06.2021
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| Online Access | Get full text |
| ISSN | 2432-1982 |
| DOI | 10.11540/bjsiam.31.2_2 |
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| Author | 池田, 正弘 |
|---|---|
| Author_xml | – sequence: 1 fullname: 池田, 正弘 organization: 理化学研究所革新知能統合研究科数理科学チーム |
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| Copyright | 2021日本応用数理学会 |
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| References | [3] Andersen, R., Chung, F., and Lang, K., Using PageRank to locally partition a graph, Internet Mathematics, 4 (2007), 35–64. [16] Lee, J. R., Gharan, S. O., and Trevisan, L., Multiway spectral partitioning and higher-order Cheeger ineuqalities, Journal of the ACM, 61 (2014), 37–30. [25] Takai, Y., Miyauchi, A., Ikeda, M., and Yoshida, Y., Hypergraph clustering based on Pagerank, 26nd ACM SIGKDD Conference on Knowledge Discovery and Data mining (KDD), 2020, 1970–1978. [14] Kwok, T. C., Lau, L. C., Lee, Y. T., Oveis Charan S., and Trevisan, L., Improved Cheegerʼs inequality: analysis of spectral partitioning algorithms through higher order spectral gap, in Proceedings of the 45 annual ACM symposium on Theory of Computing (STOC), 2013, 11–20. [5] Back, F., Learning with submodular functions:A convex optimization perspective, Foundations and Trends in Machine Learning, 6, 2013, 145–373. [18] Li, P., and Milenkovic, O., Submodular hypergraphs: p-Laplacians, Cheeger inequalities and spectral clustering, in Proceedings of Machine Learning Research, 2018, 3014–3023. [22] Raghavendra, P., and Steurer, D., Graph expansion and the unique games conjecture, in Proceedings of the 42nd ACM Annual Symposium on Theory of Computing (STOC), 2010, 755–764. [15] Komura, Y., Nonlinear semi-groups in Hilbert space, Journal of the Mathematical Society of Japan, 19 (1967), 493–507. [12] Ikeda, M., Miyauchi, A., Takai, Y., and Yoshida, Y., Finding Cheeger cuts in hypergraphs via heat equation, arXiv:1809.04396. [28] Zhou, D., Huang, J., and Schölkopf, B., Learning with hypergraphs: Clustering, classification, and embedding, Advances in Neural Information in Processing Systems 20 (NIPS) 2007, 1601–1608. [8] Chung, F., Laplacians and the Cheeger inequality for directed graphs, Annals of Combinatorics, 9 (2005), 1–19. [20] Louis, A., Hypergraph markov operators, eigenvalues and approximation algorithms, in Proceedings of the 47 annual ACM symposium on Theory of Computing (STOC), 2015, 713–722. [21] Louis, A., Raghavendra, P., Tetali, P., and Vempala, S., Many sparse cuts via higher eigenvalues, in Proceedings of the 44 annual ACM symposium on Theory of Computing (STOC), 2012, 1131–1140. [27] Yoshida, Y., Cheeger inequalities for submodular transformations, in Proceedings of the 2019 ACM-SIAM Symposium on Discrete Algorithms (SODA), 2019, 2582–2601. [29] Zien, J. Y., Schlag, M. D. F., and Chan, P. K., Multilevel spectral hypergraph partitionaing with arbitrary vertex sizes, IEEE Transactions on Pattern Analysis and Machine Intelligence 18, 9, 1999, 1389–1399. [26] Yoshida, Y., Nonlinear Laplacian for digraphs and its applications to network analysis, in Proceedings of the 9th ACM International Conference on Web Search and Data Mining (WSDM) 2016, 483–492. [13] Jeh, G., and Widom, J., Scaling personalized web search, in Proceedings of the 12th international conference on World Wide Web (WWW), 2003, 271–279. [1] Alon, N., Eigenvalues and expanders, Combinatorica, 6 (1986), 83–96. [10] Fujii, K., Soma, T., and Yoshida, Y., Polynomial-time algorithms for submodular Laplacian system, arXiv: 1803.10923. [19] Li, Y., and Zhang, Z.-L., Digraph Laplacian and the degree of asymmetry, Internet Mathematics, 8 (2012), 381–401. [23] Raghavendra, P., Steurer, D., and Tulsiani, M., Reductions between expansion problems, in Proceedings of the 27th IEEE Annual Conference on Computational Complexity (CCC), 2012, 64–73. [4] Agarwal, S., Lim, J., Z.-Manor, L., Perona, P., Kriegman D., and Belongie, S., Beyond pairwise clustering, in IEEE Conf. on Computer Vision and Pattern Recognition, 2005, 838–845. [17] Li, P., He, N., and Milenkovic, O., Quadratic decomposable submodular function minimization, Advances in Neural Information Processing Systems 31 (NeurIPS), 2018, 1414–1422. [24] 坂内健一，高井勇輝，純粋数学とAI，数理科学，7 (2020), 6–12 [9] Chung, F., The heat kernel as the pagerank of a graph, in Proceedings of the National Academy of Sciences of the United States of America, 104 (2007), 19735–19740. [7] Chan, T-H. H., Louis, A., Tang, Z.G., and Zhang, C., Spectral properties of hypergraph Laplacian and approximation algorithms, J. ACM 65, 3 (2018), 15–48. [11] Fujishige, S., Submodular functions and optimization, volume 58 of Annals of Discrete Mathematics. Elsevier, 2nd edition, 2005. [2] Alon, N., and Milman, V. D., λ1, isoperimetric inequalities for graphs, and superconcentrators, Journal of Combinatorial Theory, Series B, 38 (1985), 73–88. [6] Bauer, F., Normalized graph Laplacians for directed graphs, Linear Algebra and its Applications, 436, 2012, 4193–4222. |
| References_xml | – reference: [21] Louis, A., Raghavendra, P., Tetali, P., and Vempala, S., Many sparse cuts via higher eigenvalues, in Proceedings of the 44 annual ACM symposium on Theory of Computing (STOC), 2012, 1131–1140. – reference: [22] Raghavendra, P., and Steurer, D., Graph expansion and the unique games conjecture, in Proceedings of the 42nd ACM Annual Symposium on Theory of Computing (STOC), 2010, 755–764. – reference: [7] Chan, T-H. H., Louis, A., Tang, Z.G., and Zhang, C., Spectral properties of hypergraph Laplacian and approximation algorithms, J. ACM 65, 3 (2018), 15–48. – reference: [27] Yoshida, Y., Cheeger inequalities for submodular transformations, in Proceedings of the 2019 ACM-SIAM Symposium on Discrete Algorithms (SODA), 2019, 2582–2601. – reference: [13] Jeh, G., and Widom, J., Scaling personalized web search, in Proceedings of the 12th international conference on World Wide Web (WWW), 2003, 271–279. – reference: [23] Raghavendra, P., Steurer, D., and Tulsiani, M., Reductions between expansion problems, in Proceedings of the 27th IEEE Annual Conference on Computational Complexity (CCC), 2012, 64–73. – reference: [16] Lee, J. R., Gharan, S. O., and Trevisan, L., Multiway spectral partitioning and higher-order Cheeger ineuqalities, Journal of the ACM, 61 (2014), 37–30. – reference: [25] Takai, Y., Miyauchi, A., Ikeda, M., and Yoshida, Y., Hypergraph clustering based on Pagerank, 26nd ACM SIGKDD Conference on Knowledge Discovery and Data mining (KDD), 2020, 1970–1978. – reference: [24] 坂内健一，高井勇輝，純粋数学とAI，数理科学，7 (2020), 6–12． – reference: [5] Back, F., Learning with submodular functions:A convex optimization perspective, Foundations and Trends in Machine Learning, 6, 2013, 145–373. – reference: [17] Li, P., He, N., and Milenkovic, O., Quadratic decomposable submodular function minimization, Advances in Neural Information Processing Systems 31 (NeurIPS), 2018, 1414–1422. – reference: [11] Fujishige, S., Submodular functions and optimization, volume 58 of Annals of Discrete Mathematics. Elsevier, 2nd edition, 2005. – reference: [4] Agarwal, S., Lim, J., Z.-Manor, L., Perona, P., Kriegman D., and Belongie, S., Beyond pairwise clustering, in IEEE Conf. on Computer Vision and Pattern Recognition, 2005, 838–845. – reference: [12] Ikeda, M., Miyauchi, A., Takai, Y., and Yoshida, Y., Finding Cheeger cuts in hypergraphs via heat equation, arXiv:1809.04396. – reference: [26] Yoshida, Y., Nonlinear Laplacian for digraphs and its applications to network analysis, in Proceedings of the 9th ACM International Conference on Web Search and Data Mining (WSDM) 2016, 483–492. – reference: [10] Fujii, K., Soma, T., and Yoshida, Y., Polynomial-time algorithms for submodular Laplacian system, arXiv: 1803.10923. – reference: [28] Zhou, D., Huang, J., and Schölkopf, B., Learning with hypergraphs: Clustering, classification, and embedding, Advances in Neural Information in Processing Systems 20 (NIPS) 2007, 1601–1608. – reference: [29] Zien, J. Y., Schlag, M. D. F., and Chan, P. K., Multilevel spectral hypergraph partitionaing with arbitrary vertex sizes, IEEE Transactions on Pattern Analysis and Machine Intelligence 18, 9, 1999, 1389–1399. – reference: [3] Andersen, R., Chung, F., and Lang, K., Using PageRank to locally partition a graph, Internet Mathematics, 4 (2007), 35–64. – reference: [20] Louis, A., Hypergraph markov operators, eigenvalues and approximation algorithms, in Proceedings of the 47 annual ACM symposium on Theory of Computing (STOC), 2015, 713–722. – reference: [14] Kwok, T. C., Lau, L. C., Lee, Y. T., Oveis Charan S., and Trevisan, L., Improved Cheegerʼs inequality: analysis of spectral partitioning algorithms through higher order spectral gap, in Proceedings of the 45 annual ACM symposium on Theory of Computing (STOC), 2013, 11–20. – reference: [18] Li, P., and Milenkovic, O., Submodular hypergraphs: p-Laplacians, Cheeger inequalities and spectral clustering, in Proceedings of Machine Learning Research, 2018, 3014–3023. – reference: [8] Chung, F., Laplacians and the Cheeger inequality for directed graphs, Annals of Combinatorics, 9 (2005), 1–19. – reference: [9] Chung, F., The heat kernel as the pagerank of a graph, in Proceedings of the National Academy of Sciences of the United States of America, 104 (2007), 19735–19740. – reference: [15] Komura, Y., Nonlinear semi-groups in Hilbert space, Journal of the Mathematical Society of Japan, 19 (1967), 493–507. – reference: [1] Alon, N., Eigenvalues and expanders, Combinatorica, 6 (1986), 83–96. – reference: [6] Bauer, F., Normalized graph Laplacians for directed graphs, Linear Algebra and its Applications, 436, 2012, 4193–4222. – reference: [19] Li, Y., and Zhang, Z.-L., Digraph Laplacian and the degree of asymmetry, Internet Mathematics, 8 (2012), 381–401. – reference: [2] Alon, N., and Milman, V. D., λ1, isoperimetric inequalities for graphs, and superconcentrators, Journal of Combinatorial Theory, Series B, 38 (1985), 73–88. |
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| Title | ハイパーグラフ上の熱とそのネットワーク解析への応用 |
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