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Community Learning of Ising Models

Ilchi Ghazaan, Saeed | 2018

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 51180 (19)
  4. University: Sharif University of Technology
  5. Department: Computer Engineering
  6. Advisor(s): Motahari, Abolfazl
  7. Abstract:
  8. Ising model is a Markov Random Field (MRF) with binary random variables which has a vast literature in both theoretical and practical sides. In this thesis, we investigate two important statistical problems on this model. Learning the structure of MRFs has a long history and had a significant progress in the recent years. The goal of this problem is to find the independence graph of MRF using the samples generated from it. Specifically, we focus on the structure learning of ising models. Important algorithms for finding the structures had been reviewed. Additionally, we introduced information-theoretical and computational limitations of this problem. The second problem is community detection on Ising Block Model (IBM). IBM is introduced in 2016 by Berthet et al. to combine the concepts of Stochastic Block Model (SBM) with ising model. Inaccurately, IBM is a family of ising models which has the interaction matrix with block structure. Our goal is to find the communities or blocks by looking at the samples. This model and the results of original paper had been reviewed. In this thesis, we prove that the sample complexity for partial recovery of community detection is same as the exact recovery (ignore the logarithmic factor). We study the ground states of the free energy function for IBM when the corresponded independence graph is random, instead of the complete graph. Also, two simple algorithms for community detection is introduced and by simulation, we show that the accuracy of this algorithms is equal to the current algorithm but these algorithms are faster.Theoretical analysis for these algorithms introduced but simulations indicate that there is a gap between theoretical and simulation results. Closing this gap could be an interesting topic for further research
  9. Keywords:
  10. Markov Random Field (MRF) ; Ising model ; Community Detection ; Structural Learning ; Ising Block Model

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