Sampling theorems for two-dimensional isotropic random fields

The construction suggested by Lemma 1 enables transformation of a prof?lem of maximization of E, to a problem of maximiza-tion of E , ; hence randomness can be introduced. To summarize, given a quadratic function of the form E, or E,, it is possible to construct a neural network which will perform a random local search for the maximum. A rich class of optimization problems can be represented by quadratic functions [4]. A problem which not only is repre-sentable by a quadratic function but actually is equivalent to it is that of finding a minimum cut (MC) in a graph [4], 171. In what follows, we present the equivalence between the MC problem and neural networks (Theorem 4 and 5) and also show how neural networks relate to the directed min cut (DMC) problem (Theorem 6). To make the foregoing statements clear, let us start by defining the term cut in a graph. Definition: Let G = (V , E) be a weighted and undirected graph, with W being an n x n symmetric matrix of weights of the edges of G. Let Vi be a subset of V , and let V-, = V-Vl. The set of edges each of which is incident at one node in Vl and at one node in V _ is called a cut of the graph G. A minimum cut in a graph is a cut for which the sum of the corresponding edge weights is minimal over all Vl.