Learning Processes of Layered Neural Networks

A positive reinforcement type learning algorithm is formulated for a stochastic feed-forward neural network, and a learning equation similar to that of the Boltzmann machine algorithm is obtained. By applying a mean field approximation to the same stochastic feed-forward neural network, a deterministic analog feed-forward network is obtained and the back-propagation learning rule is re-derived

Absence of a classical long-range order in S=1/2S=1/2 Heisenberg antiferromagnet on triangular lattice

We study the quantum phase transition of an $S=1/2$ anisotropic $\alpha$ $(\equiv J_z/J_{xy})$ Heisenberg antiferromagnet on a triangular lattice. We calculate the sublattice magnetization and the long-range helical order-parameter and their Binder ratios on finite systems with $N \leq 36$ sites. The $N$ dependence of the Binder ratios reveals that the classical 120$^{\circ}$ N\'{e}el state occurs for $\alpha \lesssim 0.55$, whereas a critical collinear state occurs for $1/\alpha \lesssim 0.6$. This result is at odds with a widely-held belief that the ground state of a Heisenberg antiferromagnet is the 120$^{\circ}$ N\'{e}el state, but it also provides a possible mechanism explaining experimentally observed spin liquids.Comment: 4 pages, 7 figure

Probing a ferromagnetic critical regime using nonlinear susceptibility

The second order para-ferromagnetic phase transition in a series of amorphous alloys (Fe{_5}Co{_{50}}Ni{_{17-x}}Cr{_x}B{_{16}}Si{_{12}}) is investigated using nonlinear susceptibility. A simple molecular field treatment for the critical region shows that the third order suceptibility (chi{_3}) diverges on both sides of the transition temperature, and changes sign at T{_C}. This critical behaviour is observed experimentally in this series of amorphous ferromagnets, and the related assymptotic critical exponents are calculated. It is shown that using the proper scaling equations, all the exponents necessary for a complete characterization of the phase transition can be determined using linear and nonlinear susceptiblity measurements alone. Using meticulous nonlinear susceptibility measurements, it is shown that at times chi{_3} can be more sensitive than the linear susceptibility (chi{_1}) in unravelling the magnetism of ferromagnetic spin systems. A new technique for accurately determining T{_C} is discussed, which makes use of the functional form of chi{_3} in the critical region.Comment: 11 Figures, Submitted to Physical Review

Learning Algorithms of Multilayer Neural Networks

A positive reinforcement type learning algorithm is formulated for a stochastic feed-forward multilayer neural network, with far interlayer synaptic connections, and we obtain a learning rule similar to that of the Boltzmann machine on the same multilayer structure. By applying a mean field approximation to the stochastic feed-forward neural network, the generalized error back-propagation learning rule is derived for a deterministic analog feed-forward multilayer network with the far interlayer synaptic connections