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Open AccessArticle

Beyond the Maximum Storage Capacity Limit in Hopfield Recurrent Neural Networks

Center for Life Nanoscience, Istituto Italiano di Tecnologia, Viale Regina Elena 291, 00161 Rome, Italy
CNR NANOTEC-Institute of Nanotechnology c/o Campus Ecotekne, University of Salento, Via Monteroni, 73100 Lecce, Italy
Department of Physics, Sapienza University of Rome, Piazzale Aldo Moro 5, 00185 Rome, Italy
Author to whom correspondence should be addressed.
Entropy 2019, 21(8), 726;
Received: 3 July 2019 / Accepted: 23 July 2019 / Published: 25 July 2019
(This article belongs to the Special Issue Statistical Mechanics of Neural Networks)
PDF [574 KB, uploaded 25 July 2019]

Simple Summary


In a neural network, an autapse is a particular kind of synapse that links a neuron onto itself. Autapses are almost always not allowed neither in artificial nor in biological neural networks. Moreover, redundant or similar stored states tend to interact destructively. This paper shows how autapses together with stable state redundancy can improve the storage capacity of a recurrent neural network. Recent research shows how, in an N-node Hopfield neural network with autapses, the number of stored patterns (P) is not limited to the well known bound 0.14 N , as it is for networks without autapses. More precisely, it describes how, as the number of stored patterns increases well over the 0.14 N threshold, for P much greater than N, the retrieval error asymptotically approaches a value below the unit. Consequently, the reduction of retrieval errors allows a number of stored memories, which largely exceeds what was previously considered possible. Unfortunately, soon after, new results showed that, in the thermodynamic limit, given a network with autapses in this high-storage regime, the basin of attraction of the stored memories shrinks to a single state. This means that, for each stable state associated with a stored memory, even a single bit error in the initial pattern would lead the system to a stationary state associated with a different memory state. This thus limits the potential use of this kind of Hopfield network as an associative memory. This paper presents a strategy to overcome this limitation by improving the error correcting characteristics of the Hopfield neural network. The proposed strategy allows us to form what we call an absorbing-neighborhood of state surrounding each stored memory. An absorbing-neighborhood is a set defined by a Hamming distance surrounding a network state, which is an absorbing because, in the long-time limit, states inside it are absorbed by stable states in the set. We show that this strategy allows the network to store an exponential number of memory patterns, each surrounded with an absorbing-neighborhood with an exponentially growing size. View Full-Text
Keywords: recurrent neural networks; Hopfield neural networks; pattern storage recurrent neural networks; Hopfield neural networks; pattern storage

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Gosti, G.; Folli, V.; Leonetti, M.; Ruocco, G. Beyond the Maximum Storage Capacity Limit in Hopfield Recurrent Neural Networks. Entropy 2019, 21, 726.

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