Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 997A piecewise linear particle path contributing to the discrete Feynman propa-Fig. 6.4gator.we Wick–rotate the time variable t to imaginary values t ↦→ τ = it, therebymaking all integrals real [Reed and Simon (1975)].6.1.10 Application: Adaptive Path IntegralNow, we can extend the Feynman sum–over–histories (6.2), by adding thesynaptic–like weights w i = w i (t) into the measure D[x], to get the adaptivepath integral:Adaptive T ransition Amplitude =< B|A > w =∫Σ, D[w, x] e iS[x] , (6.5)where the adaptive measure D[w, x] is defined by the weighted product (ofdiscrete time steps)n∏D[w, x] = lim w i (t) dx i (t). (6.6)n−→∞In (6.6) the synaptic weights w i = w i (t) are updated by the unsupervisedHebbian–like learning rule [Hebb (1949)]:t=1w i (t + 1) = w i (t) + σ η (wi d(t) − w i a(t)), (6.7)where σ = σ(t), η = η(t) represent local signal and noise amplitudes,respectively, while superscripts d and a denote desired and achieved systemstates, respectively. Theoretically, equations (6.5–6.7) define an∞−dimensional complex–valued neural network. 8 Practically, in a com-8 For details on complex–valued neural networks, see e.g., complex–domain extensionof the standard backpropagation learning algorithm [Georgiou and Koutsougeras (1992);Benvenuto and Piazza (1992)].