Spectral Analysis of Neural Models with Stochastic Weights - Advanced Models of Neural Networks - page 205

Chapter 11
Spectral Analysis of Neural Models
with Stochastic Weights

Abstract Spectral analysis of neural networks with stochastic weights (stemming
from the solution of Schrödinger's diffusion equation) has shown that: (a) The
Gaussian basis functions of the weights express the distribution of the energy with
respect to the weights' value. The smaller the spread of the basis functions is, the
larger becomes the spectral (energy) content that can be captured therein. Narrow
spread of the basis functions results in wide range of frequencies of the Fourier
transformed pulse, (b) The stochastic weights satisfy an equation which is analogous
to the principle of uncertainty.

11.1 Overview

The energy spectrum of the stochastic weights that follow the quantum harmonic
oscillator (QHO) model is studied. To this end, previous results on wavelets' energy
spectrum are used [ 2 , 42 , 149 , 168 ]. Spectral analysis of the stochastic weights shows
that: (a) The Gaussian membership functions of the weights express the distribution
of energy with respect to the weights' value. The smaller the spread of the basis
functions is, the larger becomes the spectral (energy) content that can be captured
therein (b) The stochastic weights satisfy an equation which is analogous to the
principle of uncertainty. Moreover, simulation and numerical results were provided
to support the argument that the representation of the basis functions in the space
and in the frequency domain cannot be, in both cases, too sharply localized.