Hopfield network

A Hopfield network (or associative memory) is a form of recurrent neural network, or a spin glass system, that can serve as a content-addressable memory. The Hopfield network, named for John Hopfield, consists of a single layer of neurons, where each neuron is connected to every other neuron except itself. These connections are bidirectional and symmetric, meaning the weight of the connection from neuron ''i'' to neuron ''j'' is the same as the weight from neuron ''j'' to neuron ''i''. Patterns are associatively recalled by fixing certain inputs, and dynamically evolve the network to minimize an energy function, towards local energy minimum states that correspond to stored patterns. Patterns are associatively learned (or "stored") by a Hebbian learning algorithm.

One of the key features of Hopfield networks is their ability to recover complete patterns from partial or noisy inputs, making them robust in the face of incomplete or corrupted data. Their connection to statistical mechanics, recurrent networks, and human cognitive psychology has led to their application in various fields, including physics, psychology, neuroscience, and machine learning theory and practice.

History

One origin of associative memory is human cognitive psychology, specifically the associative memory. Frank Rosenblatt studied "close-loop cross-coupled perceptrons", which are 3-layered perceptron networks whose middle layer contains recurrent connections that change by a Hebbian learning rule.

Another model of associative memory is where the output does not loop back to the input. W. K. Taylor proposed such a model trained by Hebbian learning in 1956. Karl Steinbuch, who wanted to understand learning, and inspired by watching his children learn, published the Lernmatrix in 1961. It was translated to English in 1963. Similar research was done with the ''correlogram'' of D. J. Willshaw et al. in 1969. Teuvo Kohonen trained an associative memory by gradient descent in 1974.
Another origin of associative memory was statistical mechanics. The Ising model was published in 1920s as a model of magnetism, however it studied the thermal equilibrium, which does not change with time. Roy J. Glauber in 1963 studied the Ising model evolving in time, as a process towards thermal equilibrium ( Glauber dynamics), adding in the component of time.

The second component to be added was adaptation to stimulus. Described independently by Kaoru Nakano in 1971 and Shun'ichi Amari in 1972, they proposed to modify the weights of an Ising model by Hebbian learning rule as a model of associative memory. The same idea was published by in 1974, who was acknowledged by Hopfield in his 1982 paper.

See Carpenter (1989) and Cowan (1990) for a technical description of some of these early works in associative memory.

The Sherrington–Kirkpatrick model of spin glass, published in 1975, is the Hopfield network with random initialization. Sherrington and Kirkpatrick found that it is highly likely for the energy function of the SK model to have many local minima. In the 1982 paper, Hopfield applied this recently developed theory to study the Hopfield network with binary activation functions. In a 1984 paper he extended this to continuous activation functions. It became a standard model for the study of neural networks through statistical mechanics.

A major advance in memory storage capacity was developed by Dimitry Krotov and Hopfield in 2016 through a change in network dynamics and energy function. This idea was further extended by Demircigil and collaborators in 2017. The continuous dynamics of large memory capacity models was developed in a series of papers between 2016 and 2020. Large memory storage capacity Hopfield Networks are now called Dense Associative Memories or modern Hopfield networks.

In 2024, John J. Hopfield and Geoffrey E. Hinton were awarded the Nobel Prize in Physics for their foundational contributions to machine learning, such as the Hopfield network.

Structure

The units in Hopfield nets are binary threshold units, i.e. the units only take on two different values for their states, and the value is determined by whether or not the unit's input exceeds its threshold $U_i$. Discrete Hopfield nets describe relationships between binary (firing or not-firing) neurons $1,2,\ldots,i,j,\ldots,N$. At a certain time, the state of the neural net is described by a vector $V$, which records which neurons are firing in a binary word of $N$ bits.

The interactions $w_$ between neurons have units that usually take on values of 1 or −1, and this convention will be used throughout this article. However, other literature might use units that take values of 0 and 1. These interactions are "learned" via Hebb's law of association, such that, for a certain state $V^s$ and distinct nodes $i,j$

$w_ = V_i^s V_j^s$

but $w_ = 0$.

(Note that the Hebbian learning rule takes the form $w_ = (2V_i^s - 1)(2V_j^s -1)$ when the units assume values in $\$.)

Once the network is trained, $w_$ no longer evolve. If a new state of neurons $V^$ is introduced to the neural network, the net acts on neurons such that

* $V^_i \rightarrow 1$ if $\sum_j w_ V^_j \ge U_i$
* $V^_i \rightarrow -1$ if $\sum_j w_ V^_j < U_i$

where $U_i$ is the threshold value of the i'th neuron (often taken to be 0). In this way, Hopfield networks have the ability to "remember" states stored in the interaction matrix, because if a new state $V^$ is subjected to the interaction matrix, each neuron will change until it matches the original state $V^$ (see the Updates section below).

The connections in a Hopfield net typically have the following restrictions:
* $w_=0, \forall i$ (no unit has a connection with itself)
* $w_ = w_, \forall i,j$ (connections are symmetric)

The constraint that weights are symmetric guarantees that the energy function decreases monotonically while following the activation rules. A network with asymmetric weights may exhibit some periodic or chaotic behaviour; however, Hopfield found that this behavior is confined to relatively small parts of the phase space and does not impair the network's ability to act as a content-addressable associative memory system.

Hopfield also modeled neural nets for continuous values, in which the electric output of each neuron is not binary but some value between 0 and 1. He found that this type of network was also able to store and reproduce memorized states.

Notice that every pair of units ''i'' and ''j'' in a Hopfield network has a connection that is described by the connectivity weight $w_$. In this sense, the Hopfield network can be formally described as a complete undirected graph $G = \langle V, f\rangle$, where $V$ is a set of McCulloch–Pitts neurons and $f:V^2 \rightarrow \mathbb R$ is a function that links pairs of units to a real value, the connectivity weight.

Updating

Updating one unit (node in the graph simulating the artificial neuron) in the Hopfield network is performed using the following rule:

s_i \leftarrow \left\} with boundary conditions The main difference between these equations and those from the conventional feedforward networks is the presence of the second term, which is responsible for the feedback from higher layers. These top-down signals help neurons in lower layers to decide on their response to the presented stimuli. Following the general recipe it is convenient to introduce a Lagrangian function $L^A(\)$ for the $A$-th hidden layer, which depends on the activities of all the neurons in that layer. The activation functions in that layer can be defined as partial derivatives of the Lagrangian With these definitions the energy (Lyapunov) function is given by \Big \sum\limits_^ x_i^A g_i^A - L^\Big- \sum\limits_^ \sum\limits_^ \sum\limits_^ g_i^ \xi^_ g_j^A, 17 If the Lagrangian functions, or equivalently the activation functions, are chosen in such a way that the Hessians for each layer are positive semi-definite and the overall energy is bounded from below, this system is guaranteed to converge to a fixed point attractor state. The temporal derivative of this energy function is given by \tau_A \sum\limits_^ \frac \frac \frac \leq 0, 18 Thus, the hierarchical layered network is indeed an attractor network with the global energy function. This network is described by a hierarchical set of synaptic weights that can be learned for each specific problem.

See also

* Associative memory (disambiguation)
* Autoassociative memory
* Boltzmann machine – like a Hopfield net but uses annealed Gibbs sampling instead of gradient descent
* Dynamical systems model of cognition
* Ising model
* Hebbian theory

References

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External links

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Hopfield Network Javascript
– Hopfield Neural Network JAVA Applet
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{{Stochastic processes

Neural network architectures