Network inference with hidden units

Previous Article
MBE Home
This Issue
Next Article
Structural phase transitions in neural networks

2014, 11(1): 149-165. doi: 10.3934/mbe.2014.11.149

Network inference with hidden units

Joanna Tyrcha ^1, and John Hertz ^2,

1.	Department of Mathematics, Stockholm University, Kräftriket, S-106 91 Stockholm, Sweden
2.	Nordita, Stockholm University and KTH, Roslagstullsbacken 23, S-106 91 Stockholm, Sweden

Received December 2012 Revised May 2013 Published September 2013

Abstract
Related Papers

We derive learning rules for finding the connections between units in stochastic dynamical networks from the recorded history of a ``visible'' subset of the units. We consider two models. In both of them, the visible units are binary and stochastic. In one model the ``hidden'' units are continuous-valued, with sigmoidal activation functions, and in the other they are binary and stochastic like the visible ones. We derive exact learning rules for both cases. For the stochastic case, performing the exact calculation requires, in general, repeated summations over an number of configurations that grows exponentially with the size of the system and the data length, which is not feasible for large systems. We derive a mean field theory, based on a factorized ansatz for the distribution of hidden-unit states, which offers an attractive alternative for large systems. We present the results of some numerical calculations that illustrate key features of the two models and, for the stochastic case, the exact and approximate calculations.

Keywords: mean field theory, hidden units., Network inference, kinetic Ising models, latent variables.

Mathematics Subject Classification: Primary: 62M45, 82C20; Secondary: 62J0.

Citation: Joanna Tyrcha, John Hertz. Network inference with hidden units. Mathematical Biosciences & Engineering, 2014, 11 (1) : 149-165. doi: 10.3934/mbe.2014.11.149

References:

[1]	D. Ackley, G. E. Hinton and T. J. Sejnowski, A learning algorithm for Boltzmann machines, Cogn. Sci., 9 (1985), 147-169.
[2]	H. Akaike, A new look at the statistical model identification. System identification and time-series analysis, IEE Transactions on Automatic Control, AC-19 (1974), 716-723.
[3]	D. Barber, "Bayesian Reasoning and Machine Learning," chapter 11, Cambridge Univ. Press, 2012.
[4]	A. P. Dempster, N. M. Laird and D. B. Rubin, Maximum likelihood from incomplete data via the EM algorithm. With discussion, J. Roy. Stat. Soc. B, 39 (1977), 1-38.
[5]	B. Dunn and Y. Roudi, Learning and inference in a nonequilibrium Ising model with hidden nodes, Phys. Rev. E, 87 (2013), 022127. doi: 10.1103/PhysRevE.87.022127.
[6]	R. J. Glauber, Time-dependent statistics of the Ising model, J. Math. Phys., 4 (1963), 294-307. doi: 10.1063/1.1703954.
[7]	J. Hertz, Y. Roudi and J. Tyrcha, Ising models for inferring network structure from spike data, in "Principles of Neural Coding" (eds. S. Panzeri and R. R. Quiroga), CRC Press, (2013), 527-546. doi: 10.1201/b14756-31.
[8]	M. Mézard, G. Parisi and M. Virasoro, "Spin Glass Theory and Beyond," chapter 2, World Scientific Lecture Notes in Physics, 9, World Scientific Publishing Co., Inc., Teaneck, NJ, 1987.
[9]	B. A. Pearlmutter, Learning state space trajectories in recurrent neural networks, Neural Computation, 1 (1989), 263-269.
[10]	P. Peretto, Collective properties of neural networks: A statistical physics approach, Biol. Cybern., 50 (1984), 51-62. doi: 10.1007/BF00317939.
[11]	F. J. Pineda, Generalization of back-propagation to recurrent neural networks, Phys. Rev. Lett., 59 (1987), 2229-2232. doi: 10.1103/PhysRevLett.59.2229.
[12]	Y. Roudi and J. Hertz, Mean-field theory for nonequilibrium network reconstruction, Phys. Rev. Lett., 106 (2011), 048702. doi: 10.1103/PhysRevLett.106.048702.
[13]	Y. Roudi, J. Tyrcha and J. Hertz, The Ising model for neural data: Model quality and approximate methods for extracting functional connectivity, Phys. Rev. E, 79 (2009), 051915. doi: 10.1103/PhysRevE.79.051915.
[14]	D. E. Rumelhart, G. E. Hinton and R. J. Williams, Learning Internal Representations by Error Propagation, in "Parallel Distributed Processing" (eds. D. E. Rumelhart and J. L. McClelland), Vol. 1, Chapter 8, MIT Press, 1986.
[15]	L. K. Saul, T. Jaakkola and M. I. Jordan, Mean field theory for sigmoid belief networks, J. Art. Intel. Res., 4 (1996), 61-76.
[16]	E. Schneidman, M. J. Berry, R. Segev and W. Bialek, Weak pairwise correlations imply strongly correlated network states in a neural population, Nature, 440 (2006), 1007-1012. doi: 10.1038/nature04701.
[17]	G. E. Schwarz, Estimating the dimension of a model, Annals of Statistics, 6 (1978), 461-464. doi: 10.1214/aos/1176344136.
[18]	R. Sundberg, Maximum likelihood theory for incomplete data from an exponential family, Scand. J. Statistics, 1 (1974), 49-58.
[19]	D. Sherrington and S. Kirkpatrick, Solvable model of a spin-glass, Phys. Rev. Lett., 35 (1975), 1792-1796. doi: 10.1103/PhysRevLett.35.1792.
[20]	D. J. Thouless, P. W. Anderson and R. G. Palmer, Solution of "soluble model of a spin glass,'' Philos. Mag., 92 (1974), 272-279.
[21]	R. J. Williams and D. Zipser, A learning algorithm for continually running fully recurrent networks, Neural Comp., 1 (1989), 270-280. doi: 10.1162/neco.1989.1.2.270.

show all references

References:

[1]	D. Ackley, G. E. Hinton and T. J. Sejnowski, A learning algorithm for Boltzmann machines, Cogn. Sci., 9 (1985), 147-169.
[2]	H. Akaike, A new look at the statistical model identification. System identification and time-series analysis, IEE Transactions on Automatic Control, AC-19 (1974), 716-723.
[3]	D. Barber, "Bayesian Reasoning and Machine Learning," chapter 11, Cambridge Univ. Press, 2012.
[4]	A. P. Dempster, N. M. Laird and D. B. Rubin, Maximum likelihood from incomplete data via the EM algorithm. With discussion, J. Roy. Stat. Soc. B, 39 (1977), 1-38.
[5]	B. Dunn and Y. Roudi, Learning and inference in a nonequilibrium Ising model with hidden nodes, Phys. Rev. E, 87 (2013), 022127. doi: 10.1103/PhysRevE.87.022127.
[6]	R. J. Glauber, Time-dependent statistics of the Ising model, J. Math. Phys., 4 (1963), 294-307. doi: 10.1063/1.1703954.
[7]	J. Hertz, Y. Roudi and J. Tyrcha, Ising models for inferring network structure from spike data, in "Principles of Neural Coding" (eds. S. Panzeri and R. R. Quiroga), CRC Press, (2013), 527-546. doi: 10.1201/b14756-31.
[8]	M. Mézard, G. Parisi and M. Virasoro, "Spin Glass Theory and Beyond," chapter 2, World Scientific Lecture Notes in Physics, 9, World Scientific Publishing Co., Inc., Teaneck, NJ, 1987.
[9]	B. A. Pearlmutter, Learning state space trajectories in recurrent neural networks, Neural Computation, 1 (1989), 263-269.
[10]	P. Peretto, Collective properties of neural networks: A statistical physics approach, Biol. Cybern., 50 (1984), 51-62. doi: 10.1007/BF00317939.
[11]	F. J. Pineda, Generalization of back-propagation to recurrent neural networks, Phys. Rev. Lett., 59 (1987), 2229-2232. doi: 10.1103/PhysRevLett.59.2229.
[12]	Y. Roudi and J. Hertz, Mean-field theory for nonequilibrium network reconstruction, Phys. Rev. Lett., 106 (2011), 048702. doi: 10.1103/PhysRevLett.106.048702.
[13]	Y. Roudi, J. Tyrcha and J. Hertz, The Ising model for neural data: Model quality and approximate methods for extracting functional connectivity, Phys. Rev. E, 79 (2009), 051915. doi: 10.1103/PhysRevE.79.051915.
[14]	D. E. Rumelhart, G. E. Hinton and R. J. Williams, Learning Internal Representations by Error Propagation, in "Parallel Distributed Processing" (eds. D. E. Rumelhart and J. L. McClelland), Vol. 1, Chapter 8, MIT Press, 1986.
[15]	L. K. Saul, T. Jaakkola and M. I. Jordan, Mean field theory for sigmoid belief networks, J. Art. Intel. Res., 4 (1996), 61-76.
[16]	E. Schneidman, M. J. Berry, R. Segev and W. Bialek, Weak pairwise correlations imply strongly correlated network states in a neural population, Nature, 440 (2006), 1007-1012. doi: 10.1038/nature04701.
[17]	G. E. Schwarz, Estimating the dimension of a model, Annals of Statistics, 6 (1978), 461-464. doi: 10.1214/aos/1176344136.
[18]	R. Sundberg, Maximum likelihood theory for incomplete data from an exponential family, Scand. J. Statistics, 1 (1974), 49-58.
[19]	D. Sherrington and S. Kirkpatrick, Solvable model of a spin-glass, Phys. Rev. Lett., 35 (1975), 1792-1796. doi: 10.1103/PhysRevLett.35.1792.
[20]	D. J. Thouless, P. W. Anderson and R. G. Palmer, Solution of "soluble model of a spin glass,'' Philos. Mag., 92 (1974), 272-279.
[21]	R. J. Williams and D. Zipser, A learning algorithm for continually running fully recurrent networks, Neural Comp., 1 (1989), 270-280. doi: 10.1162/neco.1989.1.2.270.

[1]	Deng Lu, Maria De Iorio, Ajay Jasra, Gary L. Rosner. Bayesian inference for latent chain graphs. Foundations of Data Science, 2020, 2 (1) : 35-54. doi: 10.3934/fods.2020003
[2]	Michael Herty, Lorenzo Pareschi, Giuseppe Visconti. Mean field models for large data–clustering problems. Networks and Heterogeneous Media, 2020, 15 (3) : 463-487. doi: 10.3934/nhm.2020027
[3]	Roberto Natalini, Thierry Paul. On the mean field limit for Cucker-Smale models. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2873-2889. doi: 10.3934/dcdsb.2021164
[4]	Gerasimenko Viktor. Heisenberg picture of quantum kinetic evolution in mean-field limit. Kinetic and Related Models, 2011, 4 (1) : 385-399. doi: 10.3934/krm.2011.4.385
[5]	Michael Herty, Torsten Trimborn, Giuseppe Visconti. Mean-field and kinetic descriptions of neural differential equations. Foundations of Data Science, 2022, 4 (2) : 271-298. doi: 10.3934/fods.2022007
[6]	Xiaoyu Zheng, Peter Palffy-Muhoray. One order parameter tensor mean field theory for biaxial liquid crystals. Discrete and Continuous Dynamical Systems - B, 2011, 15 (2) : 475-490. doi: 10.3934/dcdsb.2011.15.475
[7]	Rong Yang, Li Chen. Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation. Kinetic and Related Models, 2014, 7 (2) : 381-400. doi: 10.3934/krm.2014.7.381
[8]	Joachim Crevat. Mean-field limit of a spatially-extended FitzHugh-Nagumo neural network. Kinetic and Related Models, 2019, 12 (6) : 1329-1358. doi: 10.3934/krm.2019052
[9]	Fabio Bagagiolo, Rosario Maggistro, Raffaele Pesenti. Origin-to-destination network flow with path preferences and velocity controls: A mean field game-like approach. Journal of Dynamics and Games, 2021, 8 (4) : 359-380. doi: 10.3934/jdg.2021007
[10]	Evangelos Evangelou. Approximate Bayesian inference for geostatistical generalised linear models. Foundations of Data Science, 2019, 1 (1) : 39-60. doi: 10.3934/fods.2019002
[11]	Oliver J. Maclaren, Helen M. Byrne, Alexander G. Fletcher, Philip K. Maini. Models, measurement and inference in epithelial tissue dynamics. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1321-1340. doi: 10.3934/mbe.2015.12.1321
[12]	Chjan C. Lim. Extremal free energy in a simple mean field theory for a coupled Barotropic fluid - rotating sphere system. Discrete and Continuous Dynamical Systems, 2007, 19 (2) : 361-386. doi: 10.3934/dcds.2007.19.361
[13]	Samuel N. Cohen. Uncertainty and filtering of hidden Markov models in discrete time. Probability, Uncertainty and Quantitative Risk, 2020, 5 (0) : 4-. doi: 10.1186/s41546-020-00046-x
[14]	William Chad Young, Adrian E. Raftery, Ka Yee Yeung. A posterior probability approach for gene regulatory network inference in genetic perturbation data. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1241-1251. doi: 10.3934/mbe.2016041
[15]	Franco Flandoli, Matti Leimbach. Mean field limit with proliferation. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3029-3052. doi: 10.3934/dcdsb.2016086
[16]	Yu Yang, Dongmei Xiao. Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models. Discrete and Continuous Dynamical Systems - B, 2010, 13 (1) : 195-211. doi: 10.3934/dcdsb.2010.13.195
[17]	Napoleon Bame, Samuel Bowong, Josepha Mbang, Gauthier Sallet, Jean-Jules Tewa. Global stability analysis for SEIS models with n latent classes. Mathematical Biosciences & Engineering, 2008, 5 (1) : 20-33. doi: 10.3934/mbe.2008.5.20
[18]	Darryl D. Holm, Vakhtang Putkaradze, Cesare Tronci. Collisionless kinetic theory of rolling molecules. Kinetic and Related Models, 2013, 6 (2) : 429-458. doi: 10.3934/krm.2013.6.429
[19]	Emmanuel Frénod, Mathieu Lutz. On the Geometrical Gyro-Kinetic theory. Kinetic and Related Models, 2014, 7 (4) : 621-659. doi: 10.3934/krm.2014.7.621
[20]	Dong-Mei Zhu, Wai-Ki Ching, Robert J. Elliott, Tak-Kuen Siu, Lianmin Zhang. Hidden Markov models with threshold effects and their applications to oil price forecasting. Journal of Industrial and Management Optimization, 2017, 13 (2) : 757-773. doi: 10.3934/jimo.2016045

2018 Impact Factor: 1.313

Tools

Metrics

Other articles
by authors

on AIMS
Joanna Tyrcha John Hertz
on Google Scholar
Joanna Tyrcha John Hertz

[Back to Top]