We consider the problem of neural association for a network of non-binary neurons. Here, the task is to first memorize a set of patterns using a network of neurons whose states assume values from a finite number of integer levels. Later, the same network should be able to recall previously memorized patterns from their noisy versions. Prior work in this area consider storing a finite number of purely random patterns, and have shown that the pattern retrieval capacities (maximum number of patterns that can be memorized) scale only linearly with the number of neurons in the network. In our formulation of the problem, we concentrate on exploiting redundancy and internal structure of the patterns in order to improve the pattern retrieval capacity. Our first result shows that if the given patterns have a suitable linear-algebraic structure, i.e. comprise a sub-space of the set of all possible patterns, then the pattern retrieval capacity is in fact exponential in terms of the number of neurons. The second result extends the previous finding to cases where the patterns have weak minor components, i.e. the smallest eigenvalues of the correlation matrix tend toward zero. We will use these minor components (or the basis vectors of the pattern null space) to both increase the pattern retrieval capacity and error correction capabilities. An iterative algorithm is proposed for the learning phase, and two simple neural update algorithms are presented for the recall phase. Using analytical results and simulations, we show that the proposed methods can tolerate a fair amount of errors in the input while being able to memorize an exponentially large number of patterns.
In addition to neural networks with online learning capability, offline methods have also been used to design neural associative memories. For instance, in @cite_26 the authors assume the complete set of pattern is given in advance and calculate the weight matrix using the pseudo-inverse rule @cite_22 offline. In return, this approach helps them improve the capacity of a Hopfield network to @math , under vanishing pattern error probability condition, while being able to correct of error in the recall phase. Although this is a significant improvement to the @math scaling of the pattern retrieval capacity in @cite_27 , it comes at the price of much higher computational complexity and the lack of gradual learning ability.
@cite_27 @cite_26 @cite_22
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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. > From the Publisher: This book is a comprehensive introduction to the neural network models currently under intensive study for computational applications. It is a detailed, logically-developed treatment that covers the theory and uses of collective computational networks, including associative memory, feed forward networks, and unsupervised learning. It also provides coverage of neural network applications in a variety of problems of both theoretical and practical interest.