Linear and logarithmic capacities in associative neural networks

A model of associate memory incorporating global linearity and pointwise nonlinearities in a state space of n-dimensional binary vectors is considered. Attention is focused on the ability to store a prescribed set of state vectors as attractors within the model. Within the framework of such associative nets, a specific strategy for information storage that utilizes the spectrum of a linear operator is considered in some detail. Comparisons are made between this spectral strategy and a prior scheme that utilizes the sum of Kronecker outer products of the prescribed set of state vectors, which are to function nominally as memories. The storage capacity of the spectral strategy is linear in n (the dimension of the state space under consideration), whereas an asymptotic result of n/4 log n holds for the storage capacity of the outer product scheme. Computer-simulated results show that the spectral strategy stores information more efficiently. The preprocessing costs incurred in the two algorithms are estimated, and recursive strategies are developed for their computation