On the number of spurious memories in the Hopfield model

The outer-product method for programming the Hopfield model is discussed. The method can result in many spurious stable states-exponential in the number of vectors that are to be stored-even in the case when the vectors are orthogonal

Adiabatic Quantum Optimization for Associative Memory Recall

Hopfield networks are a variant of associative memory that recall information stored in the couplings of an Ising model. Stored memories are fixed points for the network dynamics that correspond to energetic minima of the spin state. We formulate the recall of memories stored in a Hopfield network using energy minimization by adiabatic quantum optimization (AQO). Numerical simulations of the quantum dynamics allow us to quantify the AQO recall accuracy with respect to the number of stored memories and the noise in the input key. We also investigate AQO performance with respect to how memories are stored in the Ising model using different learning rules. Our results indicate that AQO performance varies strongly with learning rule due to the changes in the energy landscape. Consequently, learning rules offer indirect methods for investigating change to the computational complexity of the recall task and the computational efficiency of AQO.Comment: 22 pages, 11 figures. Updated for clarity and figures, to appear in Frontiers of Physic

Dreaming neural networks: forgetting spurious memories and reinforcing pure ones

The standard Hopfield model for associative neural networks accounts for biological Hebbian learning and acts as the harmonic oscillator for pattern recognition, however its maximal storage capacity is $\alpha \sim 0.14$, far from the theoretical bound for symmetric networks, i.e. $\alpha =1$. Inspired by sleeping and dreaming mechanisms in mammal brains, we propose an extension of this model displaying the standard on-line (awake) learning mechanism (that allows the storage of external information in terms of patterns) and an off-line (sleep) unlearning$\&$consolidating mechanism (that allows spurious-pattern removal and pure-pattern reinforcement): this obtained daily prescription is able to saturate the theoretical bound $\alpha=1$, remaining also extremely robust against thermal noise. Both neural and synaptic features are analyzed both analytically and numerically. In particular, beyond obtaining a phase diagram for neural dynamics, we focus on synaptic plasticity and we give explicit prescriptions on the temporal evolution of the synaptic matrix. We analytically prove that our algorithm makes the Hebbian kernel converge with high probability to the projection matrix built over the pure stored patterns. Furthermore, we obtain a sharp and explicit estimate for the "sleep rate" in order to ensure such a convergence. Finally, we run extensive numerical simulations (mainly Monte Carlo sampling) to check the approximations underlying the analytical investigations (e.g., we developed the whole theory at the so called replica-symmetric level, as standard in the Amit-Gutfreund-Sompolinsky reference framework) and possible finite-size effects, finding overall full agreement with the theory.Comment: 31 pages, 12 figure

Neuro-flow Dynamics and the Learning Processes

A new description of the neural activity is introduced by the neuro-flow dynamics and the extended Hebb rule. The remarkable characteristics of the neuro-flow dynamics, such as the primacy and the recency effect during awakeness or sleep, are pointed out.Comment: 8 pages ,10 Postscript figures, LaTeX file, to appear in Chaos, Solitons and Fractal

Neural Distributed Autoassociative Memories: A Survey

Introduction. Neural network models of autoassociative, distributed memory allow storage and retrieval of many items (vectors) where the number of stored items can exceed the vector dimension (the number of neurons in the network). This opens the possibility of a sublinear time search (in the number of stored items) for approximate nearest neighbors among vectors of high dimension. The purpose of this paper is to review models of autoassociative, distributed memory that can be naturally implemented by neural networks (mainly with local learning rules and iterative dynamics based on information locally available to neurons). Scope. The survey is focused mainly on the networks of Hopfield, Willshaw and Potts, that have connections between pairs of neurons and operate on sparse binary vectors. We discuss not only autoassociative memory, but also the generalization properties of these networks. We also consider neural networks with higher-order connections and networks with a bipartite graph structure for non-binary data with linear constraints. Conclusions. In conclusion we discuss the relations to similarity search, advantages and drawbacks of these techniques, and topics for further research. An interesting and still not completely resolved question is whether neural autoassociative memories can search for approximate nearest neighbors faster than other index structures for similarity search, in particular for the case of very high dimensional vectors.Comment: 31 page

Correlating matched-filter model for analysis and optimisation of neural networks

A new formalism is described for modelling neural networks by means of which a clear physical understanding of the network behaviour can be gained. In essence, the neural net is represented by an equivalent network of matched filters which is then analysed by standard correlation techniques. The procedure is demonstrated on the synchronous Little-Hopfield network. It is shown how the ability of this network to discriminate between stored binary, bipolar codes is optimised if the stored codes are chosen to be orthogonal. However, such a choice will not often be possible and so a new neural network architecture is proposed which enables the same discrimination to be obtained for arbitrary stored codes. The most efficient convergence of the synchronous Little-Hopfield net is obtained when the neurons are connected to themselves with a weight equal to the number of stored codes. The processing gain is presented for this case. The paper goes on to show how this modelling technique can be extended to analyse the behaviour of both hard and soft neural threshold responses and a novel time-dependent threshold response is described

Harmonic analysis of neural networks

Neural networks models have attracted a lot of interest in recent years mainly because there were perceived as a new idea for computing. These models can be described as a network in which every node computes a linear threshold function. One of the main difficulties in analyzing the properties of these networks is the fact that they consist of nonlinear elements. I will present a novel approach, based on harmonic analysis of Boolean functions, to analyze neural networks. In particular I will show how this technique can be applied to answer the following two fundamental questions (i) what is the computational power of a polynomial threshold element with respect to linear threshold elements? (ii) Is it possible to get exponentially many spurious memories when we use the outer-product method for programming the Hopfield model

Correlating matched-filter model for analysis and optimisation of neural networks

A new formalism is described for modelling neural networks by means of which a clear physical understanding of the network behaviour can be gained. In essence, the neural net is represented by an equivalent network of matched filters which is then analysed by standard correlation techniques. The procedure is demonstrated on the synchronous Little-Hopfield network. It is shown how the ability of this network to discriminate between stored binary, bipolar codes is optimised if the stored codes are chosen to be orthogonal. However, such a choice will not often be possible and so a new neural network architecture is proposed which enables the same discrimination to be obtained for arbitrary stored codes. The most efficient convergence of the synchronous Little-Hopfield net is obtained when the neurons are connected to themselves with a weight equal to the number of stored codes. The processing gain is presented for this case. The paper goes on to show how this modelling technique can be extended to analyse the behaviour of both hard and soft neural threshold responses and a novel time-dependent threshold response is described