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Mean-field message-passing equations in the Hopfield model and its generalizations.

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We explored mean-field equations for Hopfield networks, finding Thouless-Anderson-Palmer (TAP) equations are better for restricted Boltzmann machine learning than belief propagation. Modified TAP equations handle correlated patterns via layered graphical models.

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Area of Science:

  • Computational Neuroscience
  • Machine Learning
  • Statistical Physics

Background:

  • Restricted Boltzmann machines (RBMs) are increasingly used for deep neural network preprocessing.
  • Hopfield networks are well-understood models for studying neural network dynamics.

Purpose of the Study:

  • To analyze mean-field equations (belief propagation and Thouless-Anderson-Palmer [TAP]) within the Hopfield model.
  • To demonstrate their application as fast, iterative message-passing algorithms for computing local neuron polarizations.
  • To compare belief propagation and TAP equations for RBM learning.

Main Methods:

  • Revisiting mean-field equations (belief propagation and TAP) in the context of Hopfield networks.
  • Developing iterative message-passing algorithms for neuron polarization.
  • Analyzing equation behavior in the retrieval phase for memorized patterns.
  • Investigating modifications to TAP equations for correlated patterns.

Main Results:

  • Belief propagation equations are pattern-dependent, while TAP equations are unique, making TAP superior for RBM learning.
  • A modified TAP equation approach is proposed for correlated patterns in Hopfield models.
  • This modification relates to message passing on multi-layered graphical models, essential for general RBMs.

Conclusions:

  • TAP equations offer a more efficient and applicable method for RBM preprocessing compared to belief propagation.
  • Correlated patterns necessitate modifications to TAP equations, revealing an underlying layered graphical structure relevant to advanced RBMs.