Local Minima of a Quadratic Binary Functional with a Quasi-Hebbian Connection Matrix

The local minima of a quadratic functional depending on binary variables are discussed. An arbitrary connection matrix can be presented in the form of quasi-Hebbian expansion where each pattern is supplied with its own individual weight. For such matrices statistical physics methods allow one to derive an equation describing local minima of the functional. A model where only one weight differs from other ones is discussed in detail. In this case the equation can be solved analytically. The critical values of the weight, for which the energy landscape is reconstructed, are obtained. Obtained results are confirmed by computer simulations.