The Annals of Statistics

Phase transition in the spiked random tensor with Rademacher prior

Wei-Kuo Chen

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We consider the problem of detecting a deformation from a symmetric Gaussian random $p$-tensor $(p\geq3)$ with a rank-one spike sampled from the Rademacher prior. Recently, in Lesieur et al. (Barbier, Krzakala, Macris, Miolane and Zdeborová (2017)), it was proved that there exists a critical threshold $\beta_{p}$ so that when the signal-to-noise ratio exceeds $\beta_{p}$, one can distinguish the spiked and unspiked tensors and weakly recover the prior via the minimal mean-square-error method. On the other side, Perry, Wein and Bandeira (Perry, Wein and Bandeira (2017)) proved that there exists a $\beta_{p}'<\beta_{p}$ such that any statistical hypothesis test cannot distinguish these two tensors, in the sense that their total variation distance asymptotically vanishes, when the signa-to-noise ratio is less than $\beta_{p}'$. In this work, we show that $\beta_{p}$ is indeed the critical threshold that strictly separates the distinguishability and indistinguishability between the two tensors under the total variation distance. Our approach is based on a subtle analysis of the high temperature behavior of the pure $p$-spin model with Ising spin, arising initially from the field of spin glasses. In particular, we identify the signal-to-noise criticality $\beta_{p}$ as the critical temperature, distinguishing the high and low temperature behavior, of the Ising pure $p$-spin mean-field spin glass model.

Article information

Ann. Statist., Volume 47, Number 5 (2019), 2734-2756.

Received: December 2017
Revised: August 2018
First available in Project Euclid: 3 August 2019

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Mathematical Reviews number (MathSciNet)

Primary: 93E10: Estimation and detection [See also 60G35]
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

BBP transition signal detection Parisi formula replica symmetry breaking spin glass spiked tensor


Chen, Wei-Kuo. Phase transition in the spiked random tensor with Rademacher prior. Ann. Statist. 47 (2019), no. 5, 2734--2756. doi:10.1214/18-AOS1763.

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  • [1] Aizenman, M., Lebowitz, J. L. and Ruelle, D. (1987). Some rigorous results on the Sherrington–Kirkpatrick spin glass model. Comm. Math. Phys. 112 3–20.
  • [2] Auffinger, A. and Chen, W.-K. (2015). The Parisi formula has a unique minimizer. Comm. Math. Phys. 335 1429–1444.
  • [3] Baik, J., Ben Arous, G. and Péché, S. (2005). Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab. 33 1643–1697.
  • [4] Barbier, J., Dia, M., Macris, N. and Krzakala, F. (2016). The mutual information in random linear estimation. In 54th Annual Allerton Conference on Communication, Control, and Computing 625–632.
  • [5] Barbier, J., Dia, M., Macris, N., Krzakala, F., Lesieur, T. and Zdeborová, L. (2016). Mutual information for symmetric rank-one matrix estimation: A proof of the replica formula. Adv. Neural Inf. Process. Syst. 29 424–432.
  • [6] Barbier, J., Krzakala, F., Macris, N., Miolane, L. and Zdeborová, L. (2017). Phase transitions, optimal errors and optimality of message-passing in generalized linear models. ArXiv e-prints.
  • [7] Barbier, J. and Macris, N. (2017). The stochastic interpolation method: A simple scheme to prove replica formulas in bayesian inference. ArXiv e-prints.
  • [8] Barbier, J., Macris, N., Dia, M. and Krzakala, F. (2017). Mutual information and optimality of approximate message-passing in random linear estimation. In ArXiv E-prints.
  • [9] Barbier, J., Macris, N. and Miolane, L. (2017). The layered structure of tensor estimation and its mutual information. In 55th Annual Allerton Conference on Communication, Control, and Computing.
  • [10] Bardina, X., Márquez-Carreras, D., Rovira, C. and Tindel, S. (2004). The $p$-spin interaction model with external field. Potential Anal. 21 311–362.
  • [11] Bayati, M. and Montanari, A. (2011). The dynamics of message passing on dense graphs, with applications to compressed sensing. IEEE Trans. Inform. Theory 57 764–785.
  • [12] Ben Arous, G., Mei, S., Montanari, A. and Nica, M. (2017). The landscape of the spiked tensor model. ArXiv e-prints.
  • [13] Benaych-Georges, F. and Nadakuditi, R. R. (2011). The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices. Adv. Math. 227 494–521.
  • [14] Bovier, A., Kurkova, I. and Löwe, M. (2002). Fluctuations of the free energy in the REM and the $p$-spin SK models. Ann. Probab. 30 605–651.
  • [15] Capitaine, M., Donati-Martin, C. and Féral, D. (2009). The largest eigenvalues of finite rank deformation of large Wigner matrices: Convergence and nonuniversality of the fluctuations. Ann. Probab. 37 1–47.
  • [16] Chen, W.-K. (2014). Chaos in the mixed even-spin models. Comm. Math. Phys. 328 867–901.
  • [17] Chen, W.-K. (2017). Variational representations for the Parisi functional and the two-dimensional Guerra–Talagrand bound. Ann. Probab. 45 3929–3966.
  • [18] Chen, W.-K. (2019). Supplement to “Phase transition in the spiked random tensor with Rademacher prior.” DOI:10.1214/18-AOS1763SUPP.
  • [19] Deshpande, Y., Abbe, E. and Montanari, A. (2017). Asymptotic mutual information for the balanced binary stochastic block model. Inf. Inference 6 125–170.
  • [20] Deshpande, Y. and Montanari, A. (2014). Information-theoretically optimal sparse pca. In IEEE Internation Symposium on Information Theory. 2197–2201.
  • [21] Donnoho, D. L., Maleki, A. and Montanari, A. (2009). Message-passing algorithms for compressed sensing. Proc. Natl. Acad. Sci. USA 106 18914–18919.
  • [22] Donnoho, D. L., Maleki, A. and Montanari, A. (2010). Message-passing algorithms for compressed sensing: I. motivation and construction. In IEEE Information Theory Workshop (ITW). 115–144.
  • [23] El Alaoui, A., Krzakala, F. and Jordan, M. I. (2017). Finite-size corrections and likelihood ratio fluctuations in the spiked wigner model. ArXiv e-prints.
  • [24] Jagannath, A. and Tobasco, I. (2016). A dynamic programming approach to the Parisi functional. Proc. Amer. Math. Soc. 144 3135–3150.
  • [25] Javanmard, A. and Montanari, A. (2013). State evolution for general approximate message passing algorithms, with applications to spatial coupling. Inf. Inference 2 115–144.
  • [26] Korada, S. B. and Macris, N. (2009). Exact solution of the gauge symmetric $p$-spin glass model on a complete graph. J. Stat. Phys. 136 205–230.
  • [27] Krzakala, F., Xu, J. and Zdeborová, L. (2016). Mutual information in rank-one matrix estimation. IEEE Information Theory Workshop (ITW) 71–75.
  • [28] Lelarge, M. and Miolane, L. Fundamental limits of symmetric low-rank matrix estimation. In Proceedings of the 2017 Conference on Learning Theory. PMLR 65.
  • [29] Lesieur, T., Krzakala, F. and Zdeborová, L. (2017). Constrained low-rank matrix estimation: Phase transitions, approximate message passing and applications. J. Stat. Mech. Theory Exp. 7 073403.
  • [30] Mézard, M., Parisi, G. and Virasoro, M. A. (1987). Spin Glass Theory and Beyond. World Scientific Lecture Notes in Physics 9. World Scientific, Teaneck, NJ.
  • [31] Miolane, L. (2017). Fundamental limits of symmetric low-rank matrix estimation: The non-symmetric case. ArXiv e-prints.
  • [32] Montanari, A., Reichman, D. and Zeitouni, O. (2017). On the limitation of spectral methods: From the Gaussian hidden clique problem to rank one perturbations of Gaussian tensors. IEEE Trans. Inform. Theory 63 1572–1579.
  • [33] Montanari, A. and Richard, E. (2014). A statistical model for tensor pca. In Neural Information Processing Systems. 2897–2905.
  • [34] Montanari, A. and Richard, E. (2016). Non-negative principal component analysis: Message passing algorithms and sharp asymptotics. IEEE Trans. Inform. Theory 62 1458–1484.
  • [35] Onatski, A., Moreira, M. J. and Hallin, M. (2013). Asymptotic power of sphericity tests for high-dimensional data. Ann. Statist. 41 1204–1231.
  • [36] Panchenko, D. (2013). The Parisi ultrametricity conjecture. Ann. of Math. (2) 177 383–393.
  • [37] Panchenko, D. (2013). The Sherrington–Kirkpatrick Model. Springer Monographs in Mathematics. Springer, New York.
  • [38] Panchenko, D. (2014). The Parisi formula for mixed $p$-spin models. Ann. Probab. 42 946–958.
  • [39] Panchenko, D. (2015). The free energy in a multi-species Sherrington–Kirkpatrick model. Ann. Probab. 43 3494–3513.
  • [40] Panchenko, D. (2018). Free energy in the mixed $p$-spin models with vector spins. Ann. Probab. 46 865–896.
  • [41] Péché, S. (2006). The largest eigenvalue of small rank perturbations of Hermitian random matrices. Probab. Theory Related Fields 134 127–173.
  • [42] Perry, A., Wein, A. S. and Bandeira, A. (2017). Statistical limits of spiked tensor models. ArXiv e-prints.
  • [43] Perry, A., Wein, A. S., Bandeira, A. S. and Moitra, A. (2018). Optimality and sub-optimality of PCA I: Spiked random matrix models. Ann. Statist. 46 2416–2451.
  • [44] Reeves, G. and Pfister, H. D. (2016). The replica-symmetric prediction for compressed sensing with gaussian matrices is exact. ArXiv e-prints.
  • [45] Talagrand, M. (2006). The Parisi formula. Ann. of Math. (2) 163 221–263.
  • [46] Talagrand, M. (2011). Mean Field Models for Spin Glasses. Volume I. Basic Examples. Springer, Berlin.
  • [47] Talagrand, M. (2011). Mean Field Models for Spin Glasses. Volume II. Advanced Replica-Symmetry and Low Temperature. Springer, Heidelberg.

Supplemental materials

  • Supplement to “Phase transition in the spiked random tensor with Rademacher prior”. The proofs of Theorems 4.1, 4.2, 4.3 and Proposition 4.1 are provided in detail in the Supplementary Material [18]. In addition, the convergence of the free energies $AF_{N}$ and $L_{N}$ defined respectively by (3.1) and (3.10) are established.