A Geometrical Analysis of Associative Memory.

Abstract

A geometrical method is proposed for analyzing the properties of an autocorrelation associative memory model. From the present geometrical viewpoint, the state transition of the model is expressed as dynamics on a sphere. The method shows that there is a critical memory ratio corresponding to the memory capacity at which the characteristic of the dynamics on the sphere changes and that the stored vectors are distributed around the stored band which is the intersection between the sphere and deformed sphere defined with the square of the inner states of neurons. The stored band seperates the sphere into the upper side, where initial state vectors in recalling processes are distributed, and the lower side, where spurious stored vectors are distributed. The method gives a geometrical picture of the dynamical behaviour in recalling processes: the state vector starts from the upper side and falls into the stored vector around the stored band or into some spurious stored vector on the lower side. Based on this picture, it is explained that Morita's partial reverse method of enhancing the memory capacity makes state vectors in the lower side unstable and stored vectors around the stored band stable.