Space of interactions with definite symmetry in neural networks with biased patterns as a spin-glass problem

• Autor(a) principal: Theumann, Alba Graciela Rivas de
• Data de Publicação: 1996
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Institucional da UFRGS
• Texto Completo: http://hdl.handle.net/10183/103640
• topic: Física da matéria condensada; Redes neurais; Simetrias; Vidros de spin; Modelo de ising; Magnetização; Renormalizacao; Modelos de vidros de spin; Biofísica; Matemática; Mecânica estatística; Equilíbrio de fase

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 NΩμ 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 NΩμ formally as a functional of renormalized site dependent ‘‘propagators’’ Gij and local ‘‘magnetizations’’ mi , which are determined from a variational principle. Calculating NΩμ in the single site or Brout approximation we recover the theory of Thouless, Anderson, and Palmer (TAP), while the mi 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.