Statistical field theory for neural networks
Moritz Helias, David Dahmen
TL;DR
This paper introduces statistical field theory methods to analyze neural networks, covering probabilistic concepts, stochastic dynamics, and mean-field approximations, with applications to disordered systems and maximum entropy models.
Contribution
It provides a comprehensive, self-contained framework applying statistical field theory to neural networks, including new derivations of mean-field and beyond-mean-field theories.
Findings
Derivation of self-consistent dynamic mean-field equations.
Systematic approach to fluctuations and phase transitions.
Application to Ising models and TAP mean field theory.
Abstract
These notes attempt a self-contained introduction into statistical field theory applied to neural networks of rate units and binary spins. The presentation consists of three parts: First, the introduction of fundamental notions of probabilities, moments, cumulants, and their relation by the linked cluster theorem, of which Wick's theorem is the most important special case; followed by the diagrammatic formulation of perturbation theory, reviewed in the statistical setting. Second, dynamics described by stochastic differential equations in the Ito-formulation, treated in the Martin-Siggia-Rose-De Dominicis-Janssen path integral formalism. With concepts from disordered systems, we then study networks with random connectivity and derive their self-consistent dynamic mean-field theory, explaining the statistics of fluctuations and the emergence of different phases with regular and chaotic…
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