Analysis of neural network energy functions using standard forms

Abstract

AbstractIn this paper, we discuss a method for analyzing the energy function of a Hopfield‐type neural network. In order to analyze the energy function which solves the given minimization problem, or simply, the problem, we define the standard form of the energy function. In general, a multidimensional energy function is complex, and it is difficult to investigate the energy functions arising in practice; but when placed in the standard form, it is possible to compare and contrast the forms of the energy functions themselves. Since an orthonormal transformation will not change the form of an energy function, we can stipulate that the standard form represents identically energy functions which have the same form. Further, according to the theory associated with standard forms, it is possible to partition a general energy function according to the eigenvalues of the connection weight matrix; and if we analyze each energy function, we can investigate the properties of the actual energy function. Using this method, we analyze the energy function given by Hopfield for the “travelling salesman problem” and study how the minimization problem is realized in the energy function. Also, we study the mutual effects of a linear combination of energy functions and discuss the results.