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Computing the p-Spectral Radii of Uniform Hypergraphs with Applications

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Abstract

The p-spectral radius of a uniform hypergraph covers many important concepts, such as Lagrangian and spectral radius of the hypergraph, and is crucial for solving spectral extremal problems of hypergraphs. In this paper, we establish a spherically constrained maximization model and propose a first-order conjugate gradient algorithm to compute the p-spectral radius of a uniform hypergraph (CSRH). By the semialgebraic nature of the adjacency tensor of a uniform hypergraph, CSRH is globally convergent and obtains the global maximizer with a high probability. When computing the spectral radius of the adjacency tensor of a uniform hypergraph, CSRH outperforms existing approaches. Furthermore, CSRH is competent to calculate the p-spectral radius of a hypergraph with millions of vertices and to approximate the Lagrangian of a hypergraph. Finally, we show that the CSRH method is capable of ranking real-world data set based on solutions generated by the p-spectral radius model.

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Notes

  1. http://www.informatik.uni-trier.de/ley/db/.

  2. We would like to thank Dr. Xutao Li for providing the database.

References

  1. Absil, P.-A., Mahony, R., Andrews, B.: Convergence of the iterates of descent methods for analytic cost functions. SIAM J. Optim. 16(2), 531–547 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Agarwal, S., Lim, J., Zelnik-Manor, L., Perona, P., Kriegman, D., Belongie, S.: Beyond pairwise clustering. In: 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’05), vol 2, pp. 838–845. IEEE (2005)

  3. Bader, B.W. , Kolda, T.G. et al.: Matlab tensor toolbox version 2.6. Available online, February (2015) URL http://www.sandia.gov/~tgkolda/TensorToolbox/

  4. Bolte, J., Daniilidis, A., Lewis, A.: The Łojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems. SIAM J. Optim. 17(4), 1205–1223 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bretto, A., Gillibert, L.: Hypergraph-based image representation. In: International Workshop on Graph-Based Representations in Pattern Recognition, pp. 1–11. Springer (2005)

  6. Brown, W., Simonovits, M.: Digraph extremal problems, hypergraph extremal problems, and the densities of graph structures. Discret. Math. 48(2–3), 147–162 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  7. Caraceni, A.: Lagrangians of hypergraphs, URL http://alessandracaraceni.altervista.org/MyWordpress/wp-content/uploads/2014/05/Hypergraph_Lagrangians.pdf (2011). [Online; Accessed 26 Jan 2017]

  8. Chang, J., Chen, Y., Qi, L.: Computing eigenvalues of large scale sparse tensors arising from a hypergraph. SIAM J. Sci. Comput. 38(6), A3618–A3643 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chang, K.C., Pearson, K., Zhang, T.: Perron–Frobenius theorem for nonnegative tensors. Commun. Math. Sci. 6(2), 507–520 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, L., Han, L., Zhou, L.: Computing tensor eigenvalues via homotopy methods. SIAM J. Matrix Anal. Appl. 37(1), 290–319 (2016a)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, Y., Dai, Y.-H., Han, D.: Fiber orientation distribution estimation using a Peaceman–Rachford splitting method. SIAM J. Imaging Sci. 9(2), 573–604 (2016b)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen, Y., Qi, L., Wang, Q.: Positive semi-definiteness and sum-of-squares property of fourth order four dimensional Hankel tensors. J. Comput. Appl. Math. 302, 356–368 (2016c)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cooper, J., Dutle, A.: Spectra of uniform hypergraphs. Linear Algebra Appl. 436(9), 3268–3292 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cui, C.-F., Dai, Y.-H., Nie, J.: All real eigenvalues of symmetric tensors. SIAM J. Matrix Anal. Appl. 35(4), 1582–1601 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ding, C., He, X., Husbands, P., Zha, H. Simon, H.D.: Pagerank, hits and a unified framework for link analysis. In: Proceedings of the 25th annual international ACM SIGIR conference on Research and development in information retrieval, pp. 353–354. ACM (2002)

  16. Ding, W., Qi, L., Wei, Y.: Fast Hankel tensor-vector product and its application to exponential data fitting. Numer. Linear Algebra Appl. 22(5), 814–832 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ducournau, A., Rital, S., Bretto, A., Laget, B.: A multilevel spectral hypergraph partitioning approach for color image segmentation. In: 2009 IEEE International Conference on Signal and Image Processing Applications (ICSIPA), pp. 419–424. IEEE (2009)

  18. Ducournau, A., Bretto, A., Rital, S., Laget, B.: A reductive approach to hypergraph clustering: an application to image segmentation. Pattern Recognit. 45(7), 2788–2803 (2012)

    Article  MATH  Google Scholar 

  19. Erdös, P., Stone, A.H.: On the structure of linear graphs. Bull. Am. Math. Soc. 52, 1087–1091 (1946)

    Article  MathSciNet  MATH  Google Scholar 

  20. Frankl, P., Füredi, Z.: Extremal problems whose solutions are the blowups of the small Witt-designs. J. Comb. Theory Ser. A 52(1), 129–147 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  21. Frankl, P., Rödl, V.: Hypergraphs do not jump. Combinatorica 4(2–3), 149–159 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  22. Frankl, P., Peng, Y., Rödl, V., Talbot, J.: A note on the jumping constant conjecture of Erdős. J. Comb. Theory Ser. B 97(2), 204–216 (2007)

    Article  MATH  Google Scholar 

  23. Gunopulos, D., Mannila, H., Khardon, R., Toivonen, H.: Data mining, hypergraph transversals, and machine learning. In: Proceedings of the Sixteenth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, pp. 209–216. ACM (1997)

  24. Hager, W.W., Zhang, H.: A new conjugate gradient method with guaranteed descent and an efficient line search. SIAM J. Optim. 16(1), 170–192 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hager, W.W., Zhang, H.: A survey of nonlinear conjugate gradient methods. Pac. J. Optim. 2(1), 35–58 (2006)

    MathSciNet  MATH  Google Scholar 

  26. Hu, S., Qi, L.: The Laplacian of a uniform hypergraph. J. Comb. Optim. 29(2), 331–366 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Huang, Y., Liu, Q., Zhang, S., Metaxas, D.N.: Image retrieval via probabilistic hypergraph ranking. In: 2010 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 3376–3383. IEEE (2010)

  28. Kang, L., Nikiforov, V., Yuan, X.: The \(p\)-spectral radius of \(k\)-partite and \(k\)-chromatic uniform hypergraphs. Linear Algebra Appl. 478, 81–107 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. Karypis, G., Aggarwal, R., Kumar, V., Shekhar, S.: Multilevel hypergraph partitioning: applications in VLSI domain. IEEE T. VLSI Syst. 7(1), 69–79 (1999)

    Article  Google Scholar 

  30. Keevash, P.: Hypergraph Turán problems. Surv. combinatorics 392, 83–140 (2011)

    MATH  Google Scholar 

  31. Keevash, P., Lenz, J., Mubayi, D.: Spectral extremal problems for hypergraphs. SIAM J. Discrete Math. 28(4), 1838–1854 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  32. Klamt, S., Haus, U.-U., Theis, F.: Hypergraphs and cellular networks. PLoS Comput. Biol. 5(5), e1000385 (2009)

    Article  MathSciNet  Google Scholar 

  33. Kolda, T.G., Mayo, J.R.: Shifted power method for computing tensor eigenpairs. SIAM J. Matrix Anal. Appl. 32(4), 1095–1124 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  34. Kolda, T.G., Mayo, J.R.: An adaptive shifted power method for computing generalized tensor eigenpairs. SIAM J. Matrix Anal. Appl. 35(4), 1563–1581 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Kolda, T.G., Bader, B.W., Kenny, J.P.: Higher-order web link analysis using multilinear algebra. In: Fifth IEEE International Conference on Data Mining (ICDM’05), p. 8. IEEE (2005)

  36. Konstantinova, E.V., Skorobogatov, V.A.: Application of hypergraph theory in chemistry. Discrete Math. 235(13), 365–383 (2001). Combinatorics (Prague, 1998)

  37. Krohn-Grimberghe, A., Drumond, L., Freudenthaler, C., Schmidt-Thieme, L.: Multi-relational matrix factorization using Bayesian personalized ranking for social network data. In: Proceedings of the fifth ACM international conference on Web search and data mining, pp. 173–182. ACM (2012)

  38. Li, H., Shao, J.-Y., Qi, L.: The extremal spectral radii of \(k\)-uniform supertrees. J. Comb. Optim. 32(3), 741–764 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  39. Li, X., Hu, W., Shen, C., Dick, A., Zhang, Z.: Context-aware hypergraph construction for robust spectral clustering. IEEE T. Knowl. Data En. 26(10), 2588–2597 (2014)

    Article  Google Scholar 

  40. Liu, Y., Shao, J., Xiao, J., Wu, F., Zhuang, Y.: Hypergraph spectral hashing for image retrieval with heterogeneous social contexts. Neurocomputing 119, 49–58 (2013)

    Article  Google Scholar 

  41. Lu, L., Man, S.: Connected hypergraphs with small spectral radius. Linear Algebra Appl. 509, 206–227 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  42. Michoel, T., Nachtergaele, B.: Alignment and integration of complex networks by hypergraph-based spectral clustering. Phys. Rev. E 86(5), 056111 (2012)

    Article  Google Scholar 

  43. Motzkin, T.S., Straus, E.G.: Maxima for graphs and a new proof of a theorem of Turán. Can. J. Math. 17, 533–540 (1965)

    Article  MATH  Google Scholar 

  44. Mubayi, D.: A hypergraph extension of Turán’s theorem. J. Comb. Theory Ser. B 96(1), 122–134 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  45. Ng, M., Qi, L., Zhou, G.: Finding the largest eigenvalue of a nonnegative tensor. SIAM J. Matrix Anal. Appl. 31(3), 1090–1099 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  46. Ng, M.K.-P., Li, X., Ye, Y.: Multirank: co-ranking for objects and relations in multi-relational data. In: Proceedings of the 17th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1217–1225. ACM (2011)

  47. Nikiforov, V.: Bounds on graph eigenvalues II. Linear Algebra Appl. 427(2–3), 183–189 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  48. Nikiforov, V.: An analytic theory of extremal hypergraph problems. arXiv preprint arXiv:1305.1073, (2013)

  49. Nikiforov, V.: Analytic methods for uniform hypergraphs. Linear Algebra Appl. 457, 455–535 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  50. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer Series in Operations Research and Financial Engineering, 2nd edn. Springer, New York (2006)

    MATH  Google Scholar 

  51. Page, L., Brin, S., Motwani, R., Winograd, T.: The pagerank citation ranking: bringing order to the web. (1999)

  52. Pearson, K.J., Zhang, T.: On spectral hypergraph theory of the adjacency tensor. Graphs Comb. 30(5), 1233–1248 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  53. Peng, Y.: Using Lagrangians of hypergraphs to find non-jumping numbers I. Ann. Comb. 12(3), 307–324 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  54. Peng, Y., Zhao, C.: Generating non-jumping numbers recursively. Discret. Appl. Math. 156(10), 1856–1864 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  55. Pliakos, K., Kotropoulos, C.: Weight estimation in hypergraph learning. In: 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 1161–1165. IEEE (2015)

  56. Qi, L.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40(6), 1302–1324 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  57. Qi, L., Luo, Z.: Tensor Analysis: Spectral Theory and Special Tensors. Society for Industrial and Applied Mathematics, Philadephia (2017)

    Book  MATH  Google Scholar 

  58. Rohe, K., Chatterjee, S., Yu, B.: Spectral clustering and the high-dimensional stochastic blockmodel. Ann. Stat. 39(4), 1878–1915 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  59. Sidorenko, A.F.: The maximal number of edges in a homogeneous hypergraph containing no prohibited subgraphs. Math. Notes 41(3), 247–259 (1987)

    Article  MATH  Google Scholar 

  60. Sun, L., Ji, S. Ye, J.: Hypergraph spectral learning for multi-label classification. In: Proceedings of the 14th ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 668–676. ACM (2008)

  61. Talbot, J.: Lagrangians of hypergraphs. Comb. Probab. Comput. 11(2), 199–216 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  62. Turán, P.: Eine Extremalaufgabe aus der Graphentheorie. Mat. Fiz. Lapok 48, 436–452 (1941)

    MathSciNet  MATH  Google Scholar 

  63. Turán, P.: Research problems. MTA Mat. Kutató Int. Közl 6, 417–423 (1961)

    Google Scholar 

  64. Xie, J., Chang, A.: On the Z-eigenvalues of the adjacency tensors for uniform hypergraphs. Linear Algebra Appl. 439(8), 2195–2204 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  65. Xu, Y., Yin, W.: A block coordinate descent method for regularized multiconvex optimization with applications to nonnegative tensor factorization and completion. SIAM J. Imaging Sci. 6(3), 1758–1789 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  66. Yue, J., Zhang, L., Lu, M.: Largest adjacency, signless Laplacian, and Laplacian H-eigenvalues of loose paths. Front. Math. China 11(3), 623–645 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  67. Zhou, D., Huang, J., Schölkopf, B.: Learning with hypergraphs: clustering, classification, and embedding. In: Advances in Neural Information Processing Systems, pp. 1601–1608. (2006)

  68. Zhuang, Y., Liu, Y., Wu, F., Zhang, Y., Shao, J.: Hypergraph spectral hashing for similarity search of social image. In: Proceedings of the 19th ACM International Conference on Multimedia, pp. 1457–1460. ACM (2011)

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Authors and Affiliations

  1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, China

    Jingya Chang

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

    Jingya Chang, Weiyang Ding & Liqun Qi

  3. Department of Electronic Engineering, City University of Hong Kong, Hung Hom, Kowloon, Hong Kong

    Hong Yan

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  1. Jingya Chang
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  2. Weiyang Ding
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  3. Liqun Qi
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  4. Hong Yan
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Correspondence to Liqun Qi.

Additional information

J. Chang work was partially supported by the National Natural Science Foundation of China (Grant Nos. 11401539 and 11571178). W. Ding work was partially supported by the Hong Kong Research Grant Council (Grant No. C1007-15G). L. Qi work was partially supported by the Hong Kong Research Grant Council (Grant Nos. PolyU 501913, 15302114, 15300715, 15301716 and C1007-15G). H. Yan work was supported in part by the Hong Kong Research Grants Council (Grant No. C1007-15G).

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Chang, J., Ding, W., Qi, L. et al. Computing the p-Spectral Radii of Uniform Hypergraphs with Applications. J Sci Comput 75, 1–25 (2018). https://doi.org/10.1007/s10915-017-0520-x

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