Efficient Learning in Boltzmann Machines Using Linear Response Theory

Kappen, Hilbert J.; Rodríguez, Francisco B.

doi:10.1162/089976698300017386

Cited by 170 publications

(248 citation statements)

References 22 publications

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“…Jaakkola and Jordan [121] explored the use of mixture distributions in improving the mean field approximation. A large class of techniques, including linear response theory and the TAP method [e.g., 127,155,182,195,255,267], seek to improve the mean field approximation by introducing higher-order correction terms. Typically, the lower bound on the log partition function is not preserved by these higher-order methods.…”

Section: Example 56 (Structured Mf For Factorial Hmms)mentioning

confidence: 99%

Graphical Models, Exponential Families, and Variational Inference

Wainwright¹,

Jordan²

2007

FNT in Machine Learning

2,211

2,452

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The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building large-scale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fields, including bioinformatics, communication theory, statistical physics, combinatorial optimization, signal and image processing, information retrieval and statistical machine learning. Many problems that arise in specific instancesincluding the key problems of computing marginals and modes of probability distributions -are best studied in the general setting. Working with exponential family representations, and exploiting the conjugate duality between the cumulant function and the entropy for exponential families, we develop general variational representations of the problems of computing likelihoods, marginal probabilities and most probable configurations. We describe how a wide variety of algorithms -among them sum-product, cluster variational methods, expectation-propagation, mean field methods, max-product and linear programming relaxation, as well as conic programming relaxations -can all be understood in terms of exact or approximate forms of these variational representations. The variational approach provides a complementary alternative to Markov chain Monte Carlo as a general source of approximation methods for inference in large-scale statistical models.

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Section: Example 56 (Structured Mf For Factorial Hmms)mentioning

confidence: 99%

Graphical Models, Exponential Families, and Variational Inference

Wainwright¹,

Jordan²

2007

FNT in Machine Learning

2,211

2,452

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“…Good convergence is observed in accordance with the theoretical expectations. Introduction.-Inferring interactions between the elements of a network can be posed as an inverse problem in statistical physics in terms of either equilibrium models [1][2][3] or nonequilibrium ones. The latter has recently gained a lot of attention because of the wider generality and relevance to systems where one has data on the system over time [4,5].…”

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confidence: 99%

Maximum Likelihood Reconstruction for Ising Models with Asynchronous Updates

et al. 2013

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We describe how the couplings in an asynchronous kinetic Ising model can be inferred. We consider two cases: one in which we know both the spin history and the update times and one in which we know only the spin history. For the first case, we show that one can average over all possible choices of update times to obtain a learning rule that depends only on spin correlations and can also be derived from the equations of motion for the correlations. For the second case, the same rule can be derived within a further decoupling approximation. We study all methods numerically for fully asymmetric SherringtonKirkpatrick models, varying the data length, system size, temperature, and external field. Good convergence is observed in accordance with the theoretical expectations. Introduction.-Inferring interactions between the elements of a network can be posed as an inverse problem in statistical physics in terms of either equilibrium models [1][2][3] or nonequilibrium ones. The latter has recently gained a lot of attention because of the wider generality and relevance to systems where one has data on the system over time [4,5].In this connection, the asynchronous kinetic Ising model offers a powerful platform for theoretical insight and practical applications. Under detailed balance (symmetric couplings), it converges to the celebrated maximum entropy equilibrium Ising distribution [6]; that is, the asynchronous model includes as a subclass the Gibbs equilibrium Ising model. In many recent works, this equilibrium model is used for inferring functional connectivity and building statistical descriptions, e.g., for neuronal spike trains [2]. However, spike trains and many other real life data come in the form of time series. Since it is only under strict detailed balance that the asynchronous Ising model converges to the equilibrium Ising distribution, it is important to find the relation between the couplings found from the asynchronous model and those from the equilibrium Gibbs distribution. This becomes particularly important for analyzing data using fine time bins at which temporal correlations become important.The asynchronous Ising model is also important from another perspective. Most of the work on the subject so far has focused on models with only one type of stochastic variables. The asynchronous Ising model, however, can be viewed as a doubly stochastic model where, in addition to spin configurations, the update times of the spins are themselves stochastic variables. This differs from the

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“…A different type of mean-field approximation is the Bethe approximation which reduces to the TAP approximation at lowest order and which consists in to assume that relevant coupling J ij have locally a tree-like structure. The Bethe approximation [16] can then be used and leads to approaches referred to as pseudo-moment matching methods [9,14,10,15]. This basically leads to two, possibly different mean-field solutions to the IIP: the direct one, by using the relation valid on a tree between the joint probability and the single and pairwise marginal distributions; the indirect one also called susceptibility propagation [10] relying in fact on the relation between the inverse susceptibility matrix and the set of susceptibility coefficients attached to the links of the tree [11].…”

Section: The Inverse Problem and An Heuristic Approximate Solutionmentioning

confidence: 99%

Pairwise MRF Models Selection for Traffic Inference

Furtlehner

2013

IIS

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Efficient Learning in Boltzmann Machines Using Linear Response Theory

Cited by 170 publications

References 22 publications

Graphical Models, Exponential Families, and Variational Inference

Graphical Models, Exponential Families, and Variational Inference

Maximum Likelihood Reconstruction for Ising Models with Asynchronous Updates

Pairwise MRF Models Selection for Traffic Inference

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