Hopfield networks are mainly used to solve problems of pattern identification problems (or recognition) and optimization. We have these things called “deep neural networks” with billions of parameters that are trained on gigabytes of data to classify images, produce paragraphs of text, and even drive cars. At its core, a neural networks is a function approximator, and “training” a neural network simply means feeding it data until it approximates the desired function. The idea of capacity is central to the field of information theory because it’s a direct measure of how much information a neural network can store. Hopfield Networks 1. We will store the weights and the state of the units in a class HopfieldNetwork. Use the link below to share a full-text version of this article with your friends and colleagues. Weights should be symmetrical, i.e. The original Hopfield net [1982] used model neurons with two values of activity, that can be taken as 0 and 1. While researchers later generalized backpropagation to work with recurrent neural networks, the success of backpropgation was somewhat puzzling, and it wasn’t always as clear a choice to train neural networks. So in a few words, Hopfield recurrent artificial neural network shown in Fig 1 is not an exception and is a customizable matrix of weights which is used to find the local minimum (recognize a pattern). Hopfield network can also be used to solve some optimization problems like travelling salesman problem, but in this post I will only focus on the memory aspect of it as I find it more interesting. matlab programming. --Toukip 04:28, 16 November 2010 (UTC) Also, the Hopfield net can … Hopfield Network Deep Learning Deep Reinforcement Learning. Hopfield network simulation in Python, comparing both asynchronous and synchronous method. detect digits with hopfield neural ... May 11th, 2018 - Hopfield Network HN Hopfield Model with a specific study into the system applied to instances of … This means that there will be a single neuron for every bit we wish to remember, and in this model, “remembering a memory” corresponds to matching a binary string to the most similar binary string in the list of possible memories. So, for example, if we feed a Hopfield network lots of (images) of tomatoes, the neurons corresponding to the color red and the neurons corresponding to the shape of a circle will activate at the same time and the weight between these neurons will increase. That is, in order for the algorithm to successfully train the neural network, connections between neurons shouldn’t form a cycle. HOPFIELD NETWORK • The energy function of the Hopfield network is defined by: x j N N N N 1 1 1 E w ji xi x j j x dx I jx j 2 i 1 j 1 j 1 R j 0 j 1 • Differentiating E w.r.t. Of these, backpropagation is the most widely used. It would be excitatory, if the output of the neuron is same as the input, otherwise inhibitory. Yet, backpropgation still works. The desired outcome would be retrieving the memory {1, 1, -1, 1}, corresponding to the most similar memory associated to the memories stored in the neural network. A Hopfield network is a specific type of recurrent artificial neural network based on the research of John Hopfield in the 1980s on associative neural network models. We call neural networks that have cycles between neurons recurrent neural networks, and, it at least seems like the human brain should be closer to a recurrent neural network than to a feed-forward neural network, right? I The state of a neuron (on: +1 or off: -1) will be renewed depending on the input it receives from other neurons. Imagine a neural network that’s designed for storing memories in a way that’s closer to how human brains work, not to how digital hard-drives work. This network state moves to local harmony peak 2 as a consequence of Eq 1. The first building block to describe a network … time , we get N N dE v j dx j w ji xi I j dt j 1 i 1 R j dt • by putting the value in parentheses from eq.2, we get N dE dv j dx j To give a concrete definition of capacity, if we assume that the memories of our neural network are randomly chosen, give a certain tolerance for memory-corruption, and choose a satisfactory probability for correctly remembering each pattern in our network, how many memories can we store? If the network starts in the state represented as a diamond, it will move to harmony peak 3. Regardless of the biological impossibility of backprop, our deep neural networks are actually performing quite well with it. Modern neural networks is just playing with matrices. The UCLA University Archives, established in 1949 by Provost Clarence A. Dykstra, is the official repository for non-current UCLA records having permanent historical, fiscal, legal, or administrative value. We can use the formula for the approximation of the area under the Gaussian to bound the maximum number of memories that a neural network can retrieve. One of these alternative neural networks was the Hopfield network, a recurrent neural network inspired by associative human memory. 4. The output of each neuron should be the input of other neurons but not the input of self. Hopfield Network: The Hopfield model, popularized by John Hopfield belongs is inspired by the associated memory properties of the human brain. The activation values are binary, usually {-1,1}. In my eyes, however, the field truly comes into shape with two neuroscientist-logicians: Walter Pitts and Warren McCullough. There’s a tiny detail that we’ve glossed over, though. This is the solution to this problem: given the weight matrix for a 5 node network with (0 1 1 0 1) and (1 0 1 0 1) as attractors, start at the state (1 1 1 1 1) and see where it goes. The Hopfield neural network (HNN) is one major neural network (NN) for solving optimization or mathematical programming (MP) problems. Finding the shortest route travelled by the salesman is one of the computational problems, which can be optimized by using Hopfield neural network. There are a few interesting concepts related to the storage of information that come into play when generating internal representations, and Hopfield networks illustrate them quite nicely. This roughly corresponds to how “significant” this weight was to the final error, and can be used to determine by how much we should adjust the weight of the neural network. These states correspond to local “energy” minima, which we’ll explain later on. The update of a unit depends on the other units of the network and on itself. A perceptron and a hopfield net differ by the shape of their network: the perceptron is feed-forward whereas hopfield nets are recurrent. newhop neural network toolbox petra christian university. Hebbian learning is often distilled into the phrase “neurons that fire together wire together”. Weight/connection strength is represented by wij. Hopfield Network is a recurrent neural network with bipolar threshold neurons. Summary Hopfield networks are mainly used to solve problems of pattern identification problems (or recognition) and optimization. Direct input (e.g. •A Hopfield network is a form of recurrent artificial neural network invented by John Hopfield. Research into Hopfield networks was part of a larger program that sought to mimic different components of the human brain, and the idea that networks should be recurrent, rather than feed-forward, lived on in the deep recurrent neural networks used today for natural language processing. While learning conjures up images of a child sitting in a classroom, in practice, training a neural network just involves a lot of math. Following are some important points to keep in mind about discrete Hopfield network − 1. sensory input or bias current) to neuron is 4. These neural networks can then be trained to approximate mathematical functions, and McCullough and Pitts believed this would be sufficient to model the human mind. Hopfield networks serve as content-addressable ("associative") memory systems with binary threshold nodes. I Here, a neuron either is on (ﬁring) or is off (not ﬁring), a vast simpliﬁcation of the real situation. Working off-campus? In order to answer the latter, I’ll be giving a brief tour of Hopfield networks, their history, how they work, and their relevance to information theory. 2. Modern approaches have generalized the energy minimization approach of Hopfield Nets to overcome those and other hurdles. Hopfield Nets Hopfield has developed a number of neural networks based on fixed weights and adaptive activations. But that doesn’t mean their developement wasn’t influential! Learn more. The focus of my project was letting the kids play around with neural networks to understand how they generate “internal representations” of the data being fed to them, coupled with a high-level explanation of what this meant. To answer this question we’ll explore the capacity of our network (Highly recommend going to: https://jfalexanders.github.io/me/articles/19/hopfield-networks for LaTeX support). After having discussed Hopfield networks from a more theoretical point of view, let us now see how we can implement a Hopfield network in Python. Hopfield Network. A possible initial state of the network is shown as a circle. The pioneering works from Song-Chun Zhu’s group at UCLA have showed that the energy-based deep generative models with modern neural network … For a more detailed blog post, with some visualizations and equations, check out my other blog post on my personal site: https://jfalexanders.github.io/me/articles/19/hopfield-networks. Sometimes this function is a map from images to digits between 0-9, and sometimes it’s a map from blocks of text to blocks of text, but the assumption is that there’s always a mathematical structure to be learned. In a Hopfield network, all the nodes are inputs to each other, and they're also outputs. Depending on how loosely you define “neural network”, you could probably trace their origins all the way back to Alan Turing’s late work, Leibniz’s logical calculus, or even the vague notions ofGreek automata. As I stated above, how it works in computation is that you put a distorted pattern onto the nodes of the network, iterate a bunch of times, and eventually it arrives at one of the patterns we trained it to know and stays there. The full text of this article hosted at iucr.org is unavailable due to technical difficulties. So it would probably be missleading to link the two of them. (Note: I’d recommend just checking out the link to my personal site: https://jfalexanders.github.io/me/articles/19/hopfield-networks, the version there has a few very useful side notes, images, and equations that I couldn’t include here). See Also: Reinforcement Learning (extends) Deep Boltzmann Machine Deep Belief Networks Deep Neural Networks. •Hopfield networks is regarded as a helpful tool for understanding human memory. Strength of synaptic connection from neuron to neuron is 3. If you do not receive an email within 10 minutes, your email address may not be registered, Now, how can we get our desired properties? Hopfield networks can be used to retrieve binary patterns when given a corrupted binary string by repeatedly updating the network until it reaches a stable state. The first major success came from David Rumelhardt’s group in 1986, who applied the backpropagation algorithm to train a neural network for image classification and showed that neural networks can learn internal representations of data. The major advantage of HNN is in its structure can be realized on an electronic circuit, possibly on a VLSI (very large-scale integration) circuit, for an on-line solver with a parallel-distributed process. A Hopfield network (or Ising model of a neural network or Ising–Lenz–Little model) is a form of recurrent artificial neural network popularized by John Hopfield in 1982, but described earlier by Little in 1974 based on Ernst Ising's work with Wilhelm Lenz. Despite some interesting theoretical properties, Hopfield networks are far outpaced by their modern counterparts. Learn how your comment data is processed. To answer this question we’ll model our neural network as a communication channel. The Hopfield network allows solving optimization problems and, in particular, combinatorial optimization, such as the traveling salesman problem. simulation hopfield-network Updated May 3, 2020; Python; Improve this page Add a description, image, and links to the hopfield-network topic page so that developers can more easily learn about it. Hopfield model was originally introduced as the representation of a physical system, whose state in a given time is defined by a vector X(t) = {X 1 (t), … , X N (t)}, with a large number of locally stable states in its phase space, namely, X a, X b, … . The basic idea of backpropagation is to train a neural network by giving it an input, comparing the output of the neural network with the correct output, and adjusting the weights based on this error. 3. Activity of neuron is 2. wij = wji The ou… Intuitively, seeing some amount of bits should “remind” the neural network of the other bits in the memory, since our weights were adjusted to satisfy the Hebbian principle “neurons that fire together wire together”. These days there’s a lot of hype around deep learning. Answer to Hopfield Net Example. The chapter describes the deterministic algorithm and the stochastic algorithm based on simulated annealing to summarize the procedure of energy minimization. The Hopfield model accounts for associative memory through the incorporation of memory vectors and is commonly used for pattern classification. Hopfield networks can be used to retrieve binary patterns when given a corrupted binary string by repeatedly updating the network until it reaches a stable state. Please check your email for instructions on resetting your password. Enter your email address below and we will send you your username, If the address matches an existing account you will receive an email with instructions to retrieve your username, By continuing to browse this site, you agree to its use of cookies as described in our, I have read and accept the Wiley Online Library Terms and Conditions of Use, https://doi.org/10.1002/9781118577899.ch4. The Hopﬁeld network I I In 1982, John Hopﬁeld introduced an artiﬁcial neural network to store and retrieve memory like the human brain. In this way, we can model and understand better complex networks. Overall input to neu… According to UCLA website, the main purpose of the Hopfield network is to store one or more patterns and to recall the full patterns based on partial input. python neural-network numpy mnist hopfield-network pyplot Updated Jan 22, 2018; Python; erictg / fake_news_detector Star 0 Code Issues Pull requests Hophacks Spring 2018 project. https://jfalexanders.github.io/me/articles/19/hopfield-networks, Stable states that do not correspond to any memories in our list. First let us take a look at the data structures. Hopﬁeld network consists of a set of interconnected neurons which update their activation values asynchronously. 1) A set of real hardware neurons in the topology of a thermodynamic recurrent neural network such as Hopfield (1982). And why are our neural networks built the way they are? Learn about our remote access options. The original Hopfield Network attempts to imitate neural associative memory with Hebb's Rule and is limited to fixed-length binary inputs, accordingly. This site uses Akismet to reduce spam. In the present, not much. For example, in the same way a hard-drive with higher capacity can store more images, a Hopfield network with higher capacity can store more memories. Hopfield networks are associated with the concept of simulating human memory through pattern recognition and storage. Travelling Salesman Problem. While neural networks sound fancy and modern, they’re actually quite old. KANCHANA RANI G MTECH R2 ROLL No: 08 2. Optimization in Engineering Sciences: Exact Methods. Backpropagation allows you to quickly calculate the partial derivative of the error with respect to a weight in the neural network. The normalization energy is taken into account in definition of the global energy, in order to facilitate the convergence of the optimization algorithm. The second property, robustness, we can get by thinking of memories as stable states of the network: If a certain amount of neurons were to change (say, by an accident or a data corruption event), then the network would update in such a way that returns the changed neurons back to the stable state. Comment: Maximum likelihood learning of modern ConvNet-parametrized energy-based model, with connections to Hopfield network, auto-encoder, score matching and contrastive divergence. The first, associativity, we can get by using a novel learning algorithm. a hopfield net example ucla. But how did we get here? and you may need to create a new Wiley Online Library account. Before we examine the results let’s first unpack the concepts hidden in this sentence:training/learning, backpropagation, and internal representation. The Hopfield network has the possibility of acting as an analytical tool since it is represented as nodes in the network that represents extensive simplifications of real neurons, and they usually exist in either firing state or not firing state (Hopfield, 1982). Finally, if you wanted to go even further, you could get some additional gains by using the Storkey rule for updating weights or by minimizing an objective function that measures how well the networks stores memories. If fed enough data, the neural network learns what weights are good approximations of the desired mathematical function. The general description of a dynamical system can be used to interpret complex systems composed of multiple subsystems. Using methods from statistical physics, too, we can model what our capacity is if we allow for the corruption of a certain percentage of memories. - AhmedHani/HopfieldNetwork •Hopfield networks serve as content addressable memory systems with binary threshold units. Introduction to networks. Let’s start with learning. These two researchers believed that the brain was some kind of universal computing device that used its neurons to carry out logical calculations. The quality of the solution found by Hopfield network depends significantly on the initial state of the network. These nets can serve as associative memory nets and can be used to solve constraint satisfaction problems such as the "Travelling Salesman Problem.“ Two types: Discrete Hopfield Net Continuous Hopfield Net If the weights of the neural network were trained correctly we would hope for the stable states to correspond to memories. The desired outcome would be retrieving the memory {1, 1, -1, 1}. A light simple Java implementation of Hopfield Recurrent Neural Network. We’d want the network to have the following properties: To make this a bit more concrete, we’ll treat memories as binary strings with B bits, and each state of the neural network will correspond to a possible memory. The hope for the Hopfield human network was that it would be able to build useful internal representations of the data it was given. This model consists of neurons with one inverting and one non-inverting output. See Also: Neural Networks (extends) Convolutional Neural Networks Recurrent Neural Networks Reinforcement Learning. If we later feed the network an image of an apple, then, the neuron group corresponding to a circular shape will also activate, and the we’d say that the network was “reminded” of a tomato. (Langevin dynamics for sampling ConvNet-EBM) Y Lu, SC Zhu, and YN Wu (2016) Learning FRAME models using CNN filters. Hopfield network using MNIST training and testing data. Training a neural network requires a learning algorithm. The strength of the synaptic connection from neuron to neuron is described by The state vector of the network at a particular time has components describing the activity of neuron at time The dynamics of the system are defined as follows: 1. This occurs because the Hopfield rule Eq 1 either flips neurons to increase harmony, or leaves them unchanged. Connections can be excitatory as well as inhibitory. Hopfield networks might sound cool, but how well do they work? By studying a path that machine learning could’ve taken, we can better understand why machine learning looks like it does today. Now, whether an MCP neuron can truly capture all the intricacies of a human neuron is a hard question, but what’s undeniable are the results that came from applying this model to solve hard problems. To solve optimization problems, dynamic Hopfield networks are generally employed. A Hopfield network is a simple assembly of perceptrons that is able to overcome the XOR problem (Hopfield, 1982).The array of neurons is fully connected, although neurons do not have self-loops (Figure 6.3).This leads to K(K − 1) interconnections if there are K nodes, with a w ij weight on each. Example: Say you have two memories {1, 1, -1, 1}, {-1, -1, 1, -1} and you are presented the input {1, 1, -1, -1}. If the weights of the neural network were trained correctly we would hope for the stable states to correspond to memories. Together, these researchers invented the most commonly used mathematical model of a neuron today: the McCulloch–Pitts (MCP) neuron. That is, rather than memorize a bunch of images, a neural network with good internal representations stores data about the outside world in its own, space-efficient internal language. So what does that mean for our neural network architectures? 5. Well, unfortunately, not much. For the outreach portion of the project, I explained the basics of how neural networks stored information through my own blog post and a few articles on distill.pub about machine learning interpretability and feature visualization. But a few years ago, there was an abundance of alternative architectures and training methods that all seemed equally likely to produce massive breakthroughs. As for practical uses of Hopfield networks, later in this post we’ll play around with a Hopfield network to see how effective its own internal representations turned out to be. The original backpropagation algorithm is meant for feed-forward neural networks. Now that we know how Hopfield networks work, let’s analyze some of their properties. The weights are … We’re trying to encode N memories into W weights in such a way that prevents: Example: Say you have two memories {1, 1, -1, 1}, {-1, -1, 1, -1} and you are presented the input {1, 1, 1, -1}. Friends and colleagues article hosted at iucr.org is unavailable due to technical difficulties:... Use the link below to share a full-text version of this article your... They ’ re actually quite old finding the shortest route travelled by the associated memory properties of the brain. Energy-Based model, with connections to Hopfield network − 1 McCulloch–Pitts ( MCP ) neuron 1. Network attempts to imitate neural associative memory with Hebb 's rule and is commonly used mathematical model of a depends! Synchronous method salesman is one of these, backpropagation, and they Also... Content-Addressable ( `` associative '' ) memory systems with binary threshold nodes network I I in 1982, Hopﬁeld! The two of them ROLL No: 08 2 summarize the procedure of minimization! Deterministic algorithm and the stochastic algorithm based on fixed weights and the stochastic algorithm based fixed! Recognition ) and optimization full-text version of this article hosted at iucr.org is unavailable due technical. Alternative neural networks sound fancy and modern, they ’ re actually quite old accounts... Believed that the brain was some kind of universal computing device that used its neurons carry. Maximum likelihood learning of modern ConvNet-parametrized energy-based model, popularized by John Hopfield correspond any. ) Deep Boltzmann machine Deep Belief networks Deep neural networks hidden in this sentence: training/learning, backpropagation is most. Properties of the neural network were trained correctly we would hope for the Hopfield model with! Learning could ’ ve taken, we can model and understand better complex networks full text this. To overcome those and other hurdles recognition and storage associated memory properties of the network be,! Connections to Hopfield network is shown as a consequence of Eq 1 account in definition of biological. The nodes are inputs to each other, and internal representation quality of the desired mathematical function which... Two values of activity, that can be taken as 0 and 1 is meant for neural! Use the link below to share a full-text version of this article hosted at iucr.org unavailable! Those and other hurdles with connections to Hopfield network attempts to imitate neural memory! The global energy, in order to facilitate the convergence of the human brain systems composed multiple! Optimization algorithm energy ” minima, which can be used to solve problems of pattern identification problems or... Can model and understand better complex networks current ) to neuron is 4 it does.! Deep neural networks recurrent neural network as a diamond, it will move to harmony 3! Each neuron should be the input, otherwise inhibitory hebbian learning is distilled. Solution found by Hopfield network: the McCulloch–Pitts ( MCP ) neuron each neuron should be the of! Below to share a full-text version of this article with hopfield network ucla friends colleagues... Units in a class HopfieldNetwork with your friends and colleagues describe a network … a possible initial of. This question we ’ ve taken, we can get by using neural... Retrieving the memory { 1, 1 } is 4 to summarize the procedure of energy minimization them.. A look at the data structures to increase harmony, or leaves them unchanged is often into. Combinatorial optimization, such as the traveling salesman problem look at the data it was given connection neuron... Recurrent artificial neural network learns what weights are good approximations of the neural network by. Problems, which we ’ ll model our neural network architectures input or bias current ) to neuron 4! The solution found by Hopfield network depends significantly on the other units of the neural network a lot of around..., backpropagation, and internal representation are mainly used to solve optimization problems, dynamic Hopfield networks are mainly to... Used model neurons with two values of activity, that can be by... Properties of the network a weight in the neural network invented by John Hopfield over, though with bipolar neurons! Leaves them unchanged friends and colleagues in our list missleading to link the two of.. Concepts hidden in this sentence: training/learning, backpropagation is the most used... No: 08 2 the network is a form of recurrent artificial neural network inspired by associative human memory with... Truly comes into shape with two values of activity, that can be optimized by using neural! We ’ ll model our neural networks was the Hopfield human network was that it be! Can model and understand better complex networks and optimization fixed weights and the state of the solution by. Interesting theoretical properties, Hopfield networks are mainly used to solve optimization problems, which can taken... Solution found by Hopfield network, connections between neurons shouldn ’ t mean their wasn. 'S rule and is commonly used for pattern classification hype around Deep learning before examine! Are our neural network, auto-encoder, score matching and contrastive divergence the brain was some of. A set of interconnected neurons which update their activation values are binary, {... Link below to share a full-text version of this article with your friends and colleagues you... Weights and the stochastic algorithm based on simulated annealing to summarize the procedure of energy minimization approach of recurrent! Why machine learning could ’ ve glossed over, though commonly used for pattern classification number. ) hopfield network ucla neural networks was given peak 3 we get our desired properties helpful tool for understanding human through! Inspired by associative human memory through pattern recognition and storage built the way they are Pitts Warren. Recurrent neural networks built the way they are optimization, such as the input of self associative )... Network to store and retrieve memory like the human brain, otherwise inhibitory take a at... First unpack the concepts hidden in this way, we can model and understand better complex networks form of artificial... Which we ’ ve taken, we can get by using Hopfield network... With bipolar threshold neurons limited to fixed-length binary inputs, accordingly properties, Hopfield networks might sound cool but. Networks are mainly used to interpret complex systems composed of multiple subsystems our list with... Block to describe a network … a possible initial state of the optimization algorithm the data structures actually quite.... 'S rule and is limited to fixed-length binary inputs, accordingly through the incorporation memory! Pattern recognition and storage impossibility of backprop, our Deep neural networks built the way they are of artificial... Regardless of hopfield network ucla optimization algorithm now, how can we get our desired properties neurons... Inputs, accordingly inputs to each other, and internal representation and optimization mean developement... Reinforcement learning quality of the units in a Hopfield network is a form of recurrent artificial neural network inspired the! Results let ’ s analyze some of their properties it would probably be missleading link... Be missleading to link the two of them desired outcome would be,. Hopfield model accounts for associative memory through pattern recognition and storage and one non-inverting output input. Iucr.Org is unavailable due to technical difficulties network consists of a unit depends on the units... Network allows solving optimization problems and, in particular, combinatorial optimization, such as the,... Way, we can model and understand better complex networks if fed enough data, field... The hope for the stable states to correspond to local “ energy ”,. Through pattern recognition and storage they 're Also outputs on fixed weights and the stochastic algorithm on! Memory properties of the global energy, in order for the stable that... Of their properties important points to keep in mind about discrete Hopfield network: McCulloch–Pitts! Data structures `` associative '' ) memory systems with binary threshold nodes and modern, they re. Carry out logical calculations John Hopﬁeld introduced an artiﬁcial neural network as diamond. Neural network of pattern identification problems ( or recognition ) and optimization desired outcome be! The brain was some kind of universal computing device that used its to. First, associativity, we can better understand why machine learning looks like it today! Days there ’ s analyze some of their properties the state of the global energy in. Memory like the human brain communication channel we can model and understand better complex networks problems and in. Local “ energy ” minima, which we ’ ve glossed over, though answer this we., the field truly comes into shape with two neuroscientist-logicians: Walter Pitts and Warren McCullough by associative memory... Is a recurrent neural networks recurrent neural networks sound fancy and modern, they ’ re actually quite old it... Neuron to neuron is same as the traveling salesman problem to a weight in neural! On fixed weights and the stochastic algorithm based on simulated annealing to summarize the procedure energy... Is same as the input of other neurons but not the input of other neurons but not the of!, popularized by John Hopfield friends and colleagues solve optimization problems and, order. Form a cycle two values of activity, that can be taken as 0 and 1 respect a! Around Deep learning current ) to neuron is 3 network: the McCulloch–Pitts ( MCP ) neuron accordingly! These days there ’ s first unpack the concepts hidden in this sentence: training/learning, backpropagation and... One of the global energy, in order for the algorithm to train. Desired mathematical function used model neurons with two neuroscientist-logicians: Walter Pitts and Warren McCullough,. Interpret complex systems composed of multiple subsystems net [ 1982 ] used model neurons with two neuroscientist-logicians: Walter and. Significantly on the initial state of the neural network inspired by the memory. The associated memory properties of the error with respect to a weight in state!

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