The Annals of Statistics

Property testing in high-dimensional Ising models

Matey Neykov and Han Liu

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Abstract

This paper explores the information-theoretic limitations of graph property testing in zero-field Ising models. Instead of learning the entire graph structure, sometimes testing a basic graph property such as connectivity, cycle presence or maximum clique size is a more relevant and attainable objective. Since property testing is more fundamental than graph recovery, any necessary conditions for property testing imply corresponding conditions for graph recovery, while custom property tests can be statistically and/or computationally more efficient than graph recovery based algorithms. Understanding the statistical complexity of property testing requires the distinction of ferromagnetic (i.e., positive interactions only) and general Ising models. Using combinatorial constructs such as graph packing and strong monotonicity, we characterize how target properties affect the corresponding minimax upper and lower bounds within the realm of ferromagnets. On the other hand, by studying the detection of an antiferromagnetic (i.e., negative interactions only) Curie–Weiss model buried in Rademacher noise, we show that property testing is strictly more challenging over general Ising models. In terms of methodological development, we propose two types of correlation based tests: computationally efficient screening for ferromagnets, and score type tests for general models, including a fast cycle presence test. Our correlation screening tests match the information-theoretic bounds for property testing in ferromagnets in certain regimes.

Article information

Source
Ann. Statist., Volume 47, Number 5 (2019), 2472-2503.

Dates
Received: April 2018
First available in Project Euclid: 3 August 2019

Permanent link to this document
https://projecteuclid.org/euclid.aos/1564797854

Digital Object Identifier
doi:10.1214/18-AOS1754

Mathematical Reviews number (MathSciNet)
MR3988763

Subjects
Primary: 62F03: Hypothesis testing 62H15: Hypothesis testing

Keywords
Ising models minimax testing Dobrushin’s comparison theorem antiferromagnetic Curie–Weiss detection two-point function bounds correlation tests

Citation

Neykov, Matey; Liu, Han. Property testing in high-dimensional Ising models. Ann. Statist. 47 (2019), no. 5, 2472--2503. doi:10.1214/18-AOS1754. https://projecteuclid.org/euclid.aos/1564797854


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References

  • Addario-Berry, L., Broutin, N., Devroye, L. and Lugosi, G. (2010). On combinatorial testing problems. Ann. Statist. 38 3063–3092.
  • Ahmed, A. and Xing, E. P. (2009). Estimating time-varying networks. In Proc. Natl. Acad. Sci. USA 106 11878–11883.
  • Anandkumar, A., Tan, V. Y. F., Huang, F. and Willsky, A. S. (2012). High-dimensional structure estimation in Ising models: Local separation criterion. Ann. Statist. 40 1346–1375.
  • Arias-Castro, E., Bubeck, S. and Lugosi, G. (2012). Detection of correlations. Ann. Statist. 40 412–435.
  • Arias-Castro, E., Bubeck, S. and Lugosi, G. (2015). Detecting positive correlations in a multivariate sample. Bernoulli 21 209–241.
  • Arias-Castro, E., Bubeck, S., Lugosi, G. and Verzelen, N. (2018). Detecting Markov random fields hidden in white noise. Bernoulli 24 3628–3656.
  • Berthet, Q., Rigollet, P. and Srivastava, P. (2016). Exact recovery in the Ising blockmodel. Preprint. Available at arXiv:1612.03880.
  • Bhattacharya, B. B. and Mukherjee, S. (2018). Inference in Ising models. Bernoulli 24 493–525.
  • Bollobás, B. (2004). Extremal Graph Theory. Dover Publications, Inc., Mineola, NY.
  • Bresler, G. (2015). Efficiently learning Ising models on arbitrary graphs. In STOC’15—Proceedings of the 2015 ACM Symposium on Theory of Computing 771–782. ACM, New York.
  • Bresler, G., Gamarnik, D. and Shah, D. (2014). Structure learning of antiferromagnetic Ising models. In Advances in Neural Information Processing Systems.
  • Bresler, G., Mossel, E. and Sly, A. (2008). Reconstruction of Markov random fields from samples: Some observations and algorithms. In Approximation, Randomization and Combinatorial Optimization. Lecture Notes in Computer Science 5171 343–356. Springer, Berlin.
  • Cai, T., Liu, W. and Luo, X. (2011). A constrained $\ell_{1}$ minimization approach to sparse precision matrix estimation. J. Amer. Statist. Assoc. 106 594–607.
  • Chow, C. and Liu, C. (1968). Approximating discrete probability distributions with dependence trees. IEEE Trans. Inform. Theory 14 462–467.
  • Daskalakis, C., Dikkala, N. and Kamath, G. (2018). Testing Ising models. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms 1989–2007. SIAM, Philadelphia, PA.
  • Föllmer, H. (1988). Random fields and diffusion processes. In École D’Été de Probabilités de Saint-Flour XV–XVII, 198587. Lecture Notes in Math. 1362 101–203. Springer, Berlin.
  • Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. 6 721–741.
  • Gheissari, R., Lubetzky, E. and Peres, Y. (2017). Concentration inequalities for polynomials of contracting Ising models. Preprint. Available at arXiv:1706.00121.
  • Grimmett, G. (2018). Probability on Graphs: Random Processes on Graphs and Lattices. Institute of Mathematical Statistics Textbooks 8. Cambridge Univ. Press, Cambridge.
  • Ising, E. (1925). Beitrag zur theorie des ferromagnetismus. Z. Phys. A, Hadrons Nucl. 31 253–258.
  • Kochmański, M., Paszkiewicz, T. and Wolski, S. (2013). Curie–Weiss magnet—A simple model of phase transition. Eur. J. Phys. 34 1555.
  • Liu, H., Lafferty, J. and Wasserman, L. (2009). The nonparanormal: Semiparametric estimation of high dimensional undirected graphs. J. Mach. Learn. Res. 10 2295–2328.
  • Meinshausen, N. and Bühlmann, P. (2006). High-dimensional graphs and variable selection with the lasso. Ann. Statist. 34 1436–1462.
  • Montanari, A. and Pereira, J. A. (2009). Which graphical models are difficult to learn? In Advances in Neural Information Processing Systems.
  • Mukherjee, R., Mukherjee, S. and Yuan, M. (2018). Global testing against sparse alternatives under Ising models. Ann. Statist. 46 2062–2093.
  • Neykov, M. and Liu, H. (2019). Supplement to “Property testing in high dimensional Ising models.” DOI:10.1214/18-AOS1754SUPP.
  • Neykov, M., Lu, J. and Liu, H. (2016). Combinatorial inference for graphical models. Preprint. Available at arXiv:1608.03045.
  • Ravikumar, P., Wainwright, M. J. and Lafferty, J. D. (2010). High-dimensional Ising model selection using $\ell_{1}$-regularized logistic regression. Ann. Statist. 38 1287–1319.
  • Ravikumar, P., Wainwright, M. J., Raskutti, G. and Yu, B. (2011). High-dimensional covariance estimation by minimizing $\ell_{1}$-penalized log-determinant divergence. Electron. J. Stat. 5 935–980.
  • Rothman, A. J., Bickel, P. J., Levina, E. and Zhu, J. (2008). Sparse permutation invariant covariance estimation. Electron. J. Stat. 2 494–515.
  • Santhanam, N. P. and Wainwright, M. J. (2012). Information-theoretic limits of selecting binary graphical models in high dimensions. IEEE Trans. Inform. Theory 58 4117–4134.
  • Shanmugam, K., Tandon, R., Ravikumar, P. K. and Dimakis, A. G. (2014). On the information theoretic limits of learning Ising models. In Advances in Neural Information Processing Systems.
  • Yuan, M. and Lin, Y. (2007). Model selection and estimation in the Gaussian graphical model. Biometrika 94 19–35.

Supplemental materials

  • Supplementary Material to “Property testing in high-dimensional Ising models”. The supplement contains several auxiliary results, minimax risk lower bound proofs for ferromagnets (including that of Theorem 2.4), proofs for the correlation screening algorithm, hardness results for general Ising models and the proofs for the correlation testing algorithms for general models.