Space of interactions with definite symmetry in neural networks with biased patterns as a spin-glass problem
Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
We study the space of interactions of a connected neural network with biased patterns, when the synaptic interactions satisfy a symmetry constraint. We show that the solution to the problem requires the calculation of a quantity N⍀ analogous to the thermodynamic potential of a multiply connected Ising model with site dependent interactions, which maps the present problem into the spin-glass problem. By using a diagrammatic expansion, we express N⍀ formally as a functional of renormalized site
... renormalized site dependent "propagators" G i j and local "magnetizations" m i , which are determined from a variational principle. Calculating N⍀ in the single site or Brout approximation we recover the theory of Thouless, Anderson, and Palmer ͑TAP͒, while the m i satisfy TAP-like equations. In the impossibility of solving the equations, we analyze an approximate solution that sums only tree diagrams and interpolates between the two known results of total asymmetry, finite bias, and arbitrary symmetry with vanishing bias. The results show a small dependence on the asymmetry parameter.