Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
133 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Multi-Frequency Joint Community Detection and Phase Synchronization (2206.12276v3)

Published 16 Jun 2022 in cs.SI, cs.LG, and stat.ML

Abstract: This paper studies the joint community detection and phase synchronization problem on the \textit{stochastic block model with relative phase}, where each node is associated with an unknown phase angle. This problem, with a variety of real-world applications, aims to recover the cluster structure and associated phase angles simultaneously. We show this problem exhibits a \textit{``multi-frequency''} structure by closely examining its maximum likelihood estimation (MLE) formulation, whereas existing methods are not originated from this perspective. To this end, two simple yet efficient algorithms that leverage the MLE formulation and benefit from the information across multiple frequencies are proposed. The former is a spectral method based on the novel multi-frequency column-pivoted QR factorization. The factorization applied to the top eigenvectors of the observation matrix provides key information about the cluster structure and associated phase angles. The second approach is an iterative multi-frequency generalized power method, where each iteration updates the estimation in a matrix-multiplication-then-projection manner. Numerical experiments show that our proposed algorithms significantly improve the ability of exactly recovering the cluster structure and the accuracy of the estimated phase angles, compared to state-of-the-art algorithms.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (50)
  1. E. Abbe, “Community detection and stochastic block models: Recent developments,” The Journal of Machine Learning Research, vol. 18, no. 1, pp. 6446–6531, 2017.
  2. A. Singer, “Angular synchronization by eigenvectors and semidefinite programming,” Applied and Computational Harmonic Analysis, vol. 30, no. 1, pp. 20–36, 2011.
  3. Z. Chen, L. Li, and J. Bruna, “Supervised community detection with line graph neural networks,” in International Conference on Learning Representations, 2019.
  4. K. Berahmand, M. Mohammadi, A. Faroughi, and R. P. Mohammadiani, “A novel method of spectral clustering in attributed networks by constructing parameter-free affinity matrix,” Cluster Computing, vol. 25, no. 2, pp. 869–888, 2022.
  5. M. Girvan and M. E. Newman, “Community structure in social and biological networks,” Proceedings of the National Academy of Sciences, vol. 99, no. 12, pp. 7821–7826, 2002.
  6. K. Berahmand, A. Bouyer, and M. Vasighi, “Community detection in complex networks by detecting and expanding core nodes through extended local similarity of nodes,” IEEE Transactions on Computational Social Systems, vol. 5, no. 4, pp. 1021–1033, 2018.
  7. A. Singer, Z. Zhao, Y. Shkolnisky, and R. Hadani, “Viewing angle classification of cryo-electron microscopy images using eigenvectors,” SIAM Journal on Imaging Sciences, vol. 4, no. 2, pp. 723–759, 2011.
  8. Z. Zhao and A. Singer, “Rotationally invariant image representation for viewing direction classification in cryo-EM,” Journal of Structural Biology, vol. 186, no. 1, pp. 153–166, 2014.
  9. M. Zamiri, T. Bahraini, and H. S. Yazdi, “MVDF-RSC: Multi-view data fusion via robust spectral clustering for geo-tagged image tagging,” Expert Systems with Applications, vol. 173, p. 114657, 2021.
  10. E. Abbe, A. S. Bandeira, and G. Hall, “Exact recovery in the stochastic block model,” IEEE Transactions on Information Theory, vol. 62, no. 1, pp. 471–487, 2015.
  11. E. Abbe and C. Sandon, “Community detection in general stochastic block models: Fundamental limits and efficient algorithms for recovery,” in 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.   IEEE, 2015, pp. 670–688.
  12. E. Abbe, J. Fan, K. Wang, and Y. Zhong, “Entrywise eigenvector analysis of random matrices with low expected rank,” The Annals of Statistics, vol. 48, no. 3, p. 1452, 2020.
  13. F. Krzakala, C. Moore, E. Mossel, J. Neeman, A. Sly, L. Zdeborová, and P. Zhang, “Spectral redemption in clustering sparse networks,” Proceedings of the National Academy of Sciences, vol. 110, no. 52, pp. 20 935–20 940, 2013.
  14. L. Massoulié, “Community detection thresholds and the weak Ramanujan property,” in Proceedings of the 46th Annual ACM Symposium on Theory of Computing, 2014, pp. 694–703.
  15. A. Ng, M. Jordan, and Y. Weiss, “On spectral clustering: Analysis and an algorithm,” Advances in Neural Information Processing Systems, vol. 14, 2001.
  16. V. Vu, “A simple SVD algorithm for finding hidden partitions,” Combinatorics, Probability and Computing, vol. 27, no. 1, pp. 124–140, 2018.
  17. S.-Y. Yun and A. Proutiere, “Accurate community detection in the stochastic block model via spectral algorithms,” arXiv preprint arXiv:1412.7335, 2014.
  18. L. Su, W. Wang, and Y. Zhang, “Strong consistency of spectral clustering for stochastic block models,” IEEE Transactions on Information Theory, vol. 66, no. 1, pp. 324–338, 2019.
  19. F. McSherry, “Spectral partitioning of random graphs,” in Proceedings 42nd IEEE Symposium on Foundations of Computer Science.   IEEE, 2001, pp. 529–537.
  20. A. A. Amini and E. Levina, “On semidefinite relaxations for the block model,” The Annals of Statistics, vol. 46, no. 1, pp. 149–179, 2018.
  21. A. S. Bandeira, “Random Laplacian matrices and convex relaxations,” Foundations of Computational Mathematics, vol. 18, no. 2, pp. 345–379, 2018.
  22. O. Guédon and R. Vershynin, “Community detection in sparse networks via Grothendieck’s inequality,” Probability Theory and Related Fields, vol. 165, no. 3, pp. 1025–1049, 2016.
  23. B. Hajek, Y. Wu, and J. Xu, “Achieving exact cluster recovery threshold via semidefinite programming,” IEEE Transactions on Information Theory, vol. 62, no. 5, pp. 2788–2797, 2016.
  24. ——, “Achieving exact cluster recovery threshold via semidefinite programming: Extensions,” IEEE Transactions on Information Theory, vol. 62, no. 10, pp. 5918–5937, 2016.
  25. A. Perry and A. S. Wein, “A semidefinite program for unbalanced multisection in the stochastic block model,” in 2017 International Conference on Sampling Theory and Applications (SampTA).   IEEE, 2017, pp. 64–67.
  26. Y. Fei and Y. Chen, “Exponential error rates of SDP for block models: Beyond Grothendieck’s inequality,” IEEE Transactions on Information Theory, vol. 65, no. 1, pp. 551–571, 2018.
  27. X. Li, Y. Chen, and J. Xu, “Convex relaxation methods for community detection,” Statistical Science, vol. 36, no. 1, pp. 2–15, 2021.
  28. A. Decelle, F. Krzakala, C. Moore, and L. Zdeborová, “Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications,” Physical Review E, vol. 84, no. 6, p. 066106, 2011.
  29. Y. Chen and A. J. Goldsmith, “Information recovery from pairwise measurements,” in 2014 IEEE International Symposium on Information Theory.   IEEE, 2014, pp. 2012–2016.
  30. S. Zhang and Y. Huang, “Complex quadratic optimization and semidefinite programming,” SIAM Journal on Optimization, vol. 16, no. 3, pp. 871–890, 2006.
  31. M. Cucuringu, A. Singer, and D. Cowburn, “Eigenvector synchronization, graph rigidity and the molecule problem,” Information and Inference: A Journal of the IMA, vol. 1, no. 1, pp. 21–67, 2012.
  32. K. N. Chaudhury, Y. Khoo, and A. Singer, “Global registration of multiple point clouds using semidefinite programming,” SIAM Journal on Optimization, vol. 25, no. 1, pp. 468–501, 2015.
  33. A. S. Bandeira, C. Kennedy, and A. Singer, “Approximating the little Grothendieck problem over the orthogonal and unitary groups,” Mathematical Programming, vol. 160, no. 1, pp. 433–475, 2016.
  34. A. S. Bandeira, N. Boumal, and A. Singer, “Tightness of the maximum likelihood semidefinite relaxation for angular synchronization,” Mathematical Programming, vol. 163, no. 1, pp. 145–167, 2017.
  35. N. Boumal, “Nonconvex phase synchronization,” SIAM Journal on Optimization, vol. 26, no. 4, pp. 2355–2377, 2016.
  36. H. Liu, M.-C. Yue, and A. Man-Cho So, “On the estimation performance and convergence rate of the generalized power method for phase synchronization,” SIAM Journal on Optimization, vol. 27, no. 4, pp. 2426–2446, 2017.
  37. Y. Zhong and N. Boumal, “Near-optimal bounds for phase synchronization,” SIAM Journal on Optimization, vol. 28, no. 2, pp. 989–1016, 2018.
  38. A. S. Bandeira, Y. Chen, R. R. Lederman, and A. Singer, “Non-unique games over compact groups and orientation estimation in cryo-EM,” Inverse Problems, vol. 36, no. 6, p. 064002, 2020.
  39. A. Perry, A. S. Wein, A. S. Bandeira, and A. Moitra, “Message-passing algorithms for synchronization problems over compact groups,” Communications on Pure and Applied Mathematics, vol. 71, no. 11, pp. 2275–2322, 2018.
  40. T. Gao and Z. Zhao, “Multi-frequency phase synchronization,” in International Conference on Machine Learning.   PMLR, 2019, pp. 2132–2141.
  41. Y. Fan, Y. Khoo, and Z. Zhao, “Joint community detection and rotational synchronization via semidefinite programming,” SIAM Journal on Mathematics of Data Science, vol. 4, no. 3, pp. 1052–1081, 2022.
  42. ——, “A spectral method for joint community detection and orthogonal group synchronization,” arXiv preprint arXiv:2112.13199, 2021.
  43. S. Chen, X. Cheng, and A. M.-C. So, “Non-convex joint community detection and group synchronization via generalized power method,” arXiv preprint arXiv:2112.14204, 2021.
  44. P. Businger and G. Golub, “Linear least squares solutions by Householder transformations,” in Handbook for Automatic Computation.   Springer, 1971, pp. 111–118.
  45. P. Wang, H. Liu, Z. Zhou, and A. M.-C. So, “Optimal non-convex exact recovery in stochastic block model via projected power method,” in International Conference on Machine Learning.   PMLR, 2021, pp. 10 828–10 838.
  46. T. Tokuyama and J. Nakano, “Geometric algorithms for the minimum cost assignment problem,” Random Structures & Algorithms, vol. 6, no. 4, pp. 393–406, 1995.
  47. K. Numata and T. Tokuyama, “Splitting a configuration in a simplex,” Algorithmica, vol. 9, no. 6, pp. 649–668, 1993.
  48. A. Damle, V. Minden, and L. Ying, “Simple, direct and efficient multi-way spectral clustering,” Information and Inference: A Journal of the IMA, vol. 8, no. 1, pp. 181–203, 2019.
  49. G. W. Stewart, “A Krylov–Schur algorithm for large eigenproblems,” SIAM Journal on Matrix Analysis and Applications, vol. 23, no. 3, pp. 601–614, 2002.
  50. J. Leskovec, K. J. Lang, A. Dasgupta, and M. W. Mahoney, “Statistical properties of community structure in large social and information networks,” in Proceedings of the 17th International Conference on World Wide Web, 2008, pp. 695–704.
Citations (2)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now