Exact recovery in block spin Ising models at the critical line.(English)Zbl 1439.62254

Summary: We show how to exactly reconstruct the block structure at the critical line in the so-called Ising block model. This model was recently re-introduced by Q. Berthet et al. [Ann. Stat. 47, No. 4, 1805–1834 (2019; Zbl 1420.62268)]. There the authors show how to exactly reconstruct blocks away from the critical line and they give an upper and a lower bound on the number of observations one needs; thereby they establish a minimax optimal rate (up to constants). Our technique relies on a combination of their methods with fluctuation results obtained in [M. Löwe and K. Schubert, Electron. Commun. Probab. 23, Paper No. 53, 12 p. (2018; Zbl 1402.60027)]. The latter are extended to the full critical regime. We find that the number of necessary observations depends on whether the interaction parameter between two blocks is positive or negative: In the first case, there are about $$N\log N$$ observations required to exactly recover the block structure, while in the latter case $$\sqrt{N}\log N$$ observations suffice.

MSC:

 62P35 Applications of statistics to physics 62K10 Statistical block designs 82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics 80A10 Classical and relativistic thermodynamics

Citations:

Zbl 1420.62268; Zbl 1402.60027
Full Text:

References:

 [1] A. A. Amini and E. Levina. On semidefinite relaxations for the block model., Ann. Statist., 46(1):149-179, 2018. · Zbl 1393.62021 [2] Q. Berthet, P. Rigollet, and P. Srivastava. Exact recovery in the ising blockmodel., The Annals of Statistics, 47(4) :1805-1834, 2019. · Zbl 1420.62268 [3] A. Bovier and V. Gayrard. Rigorous bounds on the storage capacity of the dilute Hopfield model., J. Statist. Phys., 69(3-4):597-627, 1992. · Zbl 0900.82064 [4] G. Bresler. Efficiently learning Ising models on arbitrary graphs [extended abstract]. In, STOC’15—Proceedings of the 2015 ACM Symposium on Theory of Computing, pages 771-782. ACM, New York, 2015. · Zbl 1321.68397 [5] G. Bresler, E. Mossel, and A. Sly. Reconstruction of Markov random fields from samples: some observations and algorithms., SIAM J. Comput., 42(2):563-578, 2013. · Zbl 1271.68239 [6] F. Collet. Macroscopic limit of a bipartite Curie-Weiss model: a dynamical approach., J. Stat. Phys., 157(6) :1301-1319, 2014. · Zbl 1310.82020 [7] R. Cont and M. Löwe. Social distance, heterogeneity and social interactions., J. Math. Econom., 46(4):572-590, 2010. · Zbl 1232.91216 [8] R. S. Ellis., Entropy, large deviations, and statistical mechanics. Classics in Mathematics. Springer-Verlag, Berlin, 2006. Reprint of the 1985 original. · Zbl 1102.60087 [9] M. Fedele and P. Contucci. Scaling limits for multi-species statistical mechanics mean-field models., J. Stat. Phys., 144(6) :1186-1205, 2011. · Zbl 1230.82013 [10] I. Gallo, A. Barra, and P. Contucci. Parameter evaluation of a simple mean-field model of social interaction., Math. Models Methods Appl. Sci., 19(suppl.) :1427-1439, 2009. · Zbl 1175.00049 [11] I. Gallo and P. Contucci. Bipartite mean field spin systems. Existence and solution., Math. Phys. Electron. J., 14:Paper 1, 21, 2008. · Zbl 1143.82304 [12] C. Gao, Z. Ma, A. Y. Zhang, and H. H. Zhou. Achieving optimal misclassification proportion in stochastic block models., J. Mach. Learn. Res., 18:Paper No. 60, 45, 2017. · Zbl 1440.62244 [13] B. Gentz and M. Löwe. The fluctuations of the overlap in the Hopfield model with finitely many patterns at the critical temperature., Probab. Theory Related Fields, 115(3):357-381, 1999. · Zbl 0953.60094 [14] W. Kirsch. A survey on the method of moments., https://www.fernuni-hagen.de/stochastik/downloads/momente.pdf, 2015. [15] W. Kirsch and G. Toth. Two groups in a Curie-Weiss model., preprint, arXiv:1712.08477, 2017. [16] W. Kirsch and G. Toth. Critical regime in a Curie-Weiss model with two groups and heterogeneous coupling., preprint, arXiv:1807.05020, 2018. [17] W. Kirsch and G. Toth. Two groups in a Curie-Weiss model with heterogeneous coupling., preprint, arXiv:1806.06708, 2018. [18] H. Knöpfel and M. Löwe. Zur Meinungsbildung in einer heterogenen Bevölkerung—ein neuer Zugang zum Hopfield Modell., Math. Semesterber., 56(1):15-38, 2009. · Zbl 1193.91123 [19] H. Knöpfel, M. Löwe, K. Schubert, and A. Sinulis. Fluctuation results for general block spin Ising models., preprint, arXiv:1902.02080, 2019. [20] M. Löwe and K. Schubert. Fluctuations for block spin ising models., Electron. Commun. Probab., 23:12 pp., 2018. · Zbl 1402.60027 [21] M. Löwe, K. Schubert, and F. Vermet. Multi-group binary choice with social interaction and a random communication structure – a random graph approach., preprint, arXiv:1904.11890, 2019. [22] E. Mossel, J. Neeman, and A. Sly. Belief propagation, robust reconstruction and optimal recovery of block models., Ann. Appl. Probab., 26(4) :2211-2256, 2016. · Zbl 1350.05154 [23] A. A. Opoku, K. Owusu Edusei, and R. K. Ansah. A conditional Curie-Weiss model for stylized multi-group binary choice with social interaction., J. Stat. Phys., 171(1):106-126, 2018. · Zbl 1392.82078
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.