Continuous Boltzmann machine with rotor neurons

7 Acknowledgments We want to thank specially Stefan Mießbach for numerous contributions in proving the convergence properties of the net dynamic and for advice concerning the " cornered rat " example. We are very grateful to Ingrid Gabler for supplying the experimental data for this same example. 6.3 Convergence of the dynamic We show now the local convergence of the dynamic defined by the MF equations (21) and (22). We remark that this iterative update algorithm for solving equations (21) and (22) can also be viewed as discrete time integration of equation (35) with time scale. Beside of this, under condition of finite temperature and properly bounded connection weights the local convergence of the algorithm can be shown explicitly. According to the Banach fixed-point theorem local convergence is guaranteed if (49) (50) The forth rank tensor can be written as (38) substituting V by U and G by F. Since the corresponding conditions (44) and (47) still hold, the inequalities (39) and (40) are also valid for this tensor. It is thus positive definite, with positive eigenvalues (51) Now we can bound the norm as (52) (53) where 1/d is the maximal slope of F at the zero point. At the end we get from (49) condition for the local convergence: (54) V i t 1 + () f 1 T-W ij V j t () ⋅ j ∑ –     = ∆t 1 = V jl ∂ ∂f ik 1 < V jl ∂ ∂f ik