# Artificial neural network

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This article's lead section may not adequately summarize key points of its contents.  (September 2012)

An artificial neural network is an interconnected group of nodes, akin to the vast network of neurons in a brain. Here, each circular node represents an artificial neuron and an arrow represents a connection from the output of one neuron to the input of another.

In machine learning and computational neuroscience, an artificial neural network, often just named a neural network, is a mathematical model inspired by biological neural networks. A neural network consists of an interconnected group of artificial neurons, and it processes information using a connectionist approach to computation. In most cases a neural network is an adaptive system changing its structure during a learning phase. Neural networks are used for modeling complex relationships between inputs and outputs or to find patterns in data.

## Background

The inspiration for neural networks came from examination of central nervous systems. In an artificial neural network, simple artificial nodes, called "neurons", "neurodes", "processing elements" or "units", are connected together to form a network which mimics a biological neural network.

There is no single formal definition of what an artificial neural network is. Generally, it involves a network of simple processing elements exhibiting complex global behavior determined by the connections between the processing elements and element parameters. Artificial neural networks are used with algorithms designed to alter the strength of the connections in the network to produce a desired signal flow.

Neural networks are also similar to biological neural networks in performing functions collectively and in parallel by the units, rather than there being a clear delineation of subtasks to which various units are assigned. The term "neural network" usually refers to models employed in statistics, cognitive psychology and artificial intelligence. Neural network models which emulate the central nervous system are part of theoretical neuroscience and computational neuroscience.

In modern software implementations of artificial neural networks, the approach inspired by biology has been largely abandoned for a more practical approach based on statistics and signal processing. In some of these systems, neural networks or parts of neural networks (like artificial neurons) form components in larger systems that combine both adaptive and non-adaptive elements. While the more general approach of such adaptive systems is more suitable for real-world problem solving, it has far less to do with the traditional artificial intelligence connectionist models. What they do have in common, however, is the principle of non-linear, distributed, parallel and local processing and adaptation. Historically, the use of neural networks models marked a paradigm shift in the late eighties from high-level (symbolic) artificial intelligence, characterized by expert systems with knowledge embodied in if-then rules, to low-level (sub-symbolic) machine learning, characterized by knowledge embodied in the parameters of a dynamical system.

## Models

Neural network models in artificial intelligence are usually referred to as artificial neural networks (ANNs); these are essentially simple mathematical models defining a function $\scriptstyle f : X \rightarrow Y$ or a distribution over $\scriptstyle X$ or both $\scriptstyle X$ and $\scriptstyle Y$, but sometimes models are also intimately associated with a particular learning algorithm or learning rule. A common use of the phrase ANN model really means the definition of a class of such functions (where members of the class are obtained by varying parameters, connection weights, or specifics of the architecture such as the number of neurons or their connectivity).

### Network function

See also: Graphical models

The word network in the term 'artificial neural network' refers to the inter–connections between the neurons in the different layers of each system. An example system has three layers. The first layer has input neurons, which send data via synapses to the second layer of neurons, and then via more synapses to the third layer of output neurons. More complex systems will have more layers of neurons with some having increased layers of input neurons and output neurons. The synapses store parameters called "weights" that manipulate the data in the calculations.

An ANN is typically defined by three types of parameters:

1. The interconnection pattern between different layers of neurons
2. The learning process for updating the weights of the interconnections
3. The activation function that converts a neuron's weighted input to its output activation.

Mathematically, a neuron's network function $\scriptstyle f(x)$ is defined as a composition of other functions $\scriptstyle g_i(x)$, which can further be defined as a composition of other functions. This can be conveniently represented as a network structure, with arrows depicting the dependencies between variables. A widely used type of composition is the nonlinear weighted sum, where $\scriptstyle f (x) = K \left(\sum_i w_i g_i(x)\right)$, where $\scriptstyle K$ (commonly referred to as the activation function[1]) is some predefined function, such as the hyperbolic tangent. It will be convenient for the following to refer to a collection of functions $\scriptstyle g_i$ as simply a vector $\scriptstyle g = (g_1, g_2, \ldots, g_n)$.

ANN dependency graph

This figure depicts such a decomposition of $\scriptstyle f$, with dependencies between variables indicated by arrows. These can be interpreted in two ways.

The first view is the functional view: the input $\scriptstyle x$ is transformed into a 3-dimensional vector $\scriptstyle h$, which is then transformed into a 2-dimensional vector $\scriptstyle g$, which is finally transformed into $\scriptstyle f$. This view is most commonly encountered in the context of optimization.

The second view is the probabilistic view: the random variable $\scriptstyle F = f(G)$ depends upon the random variable $\scriptstyle G = g(H)$, which depends upon $\scriptstyle H=h(X)$, which depends upon the random variable $\scriptstyle X$. This view is most commonly encountered in the context of graphical models.

The two views are largely equivalent. In either case, for this particular network architecture, the components of individual layers are independent of each other (e.g., the components of $\scriptstyle g$ are independent of each other given their input $\scriptstyle h$). This naturally enables a degree of parallelism in the implementation.

Two separate depictions of the recurrent ANN dependency graph

Networks such as the previous one are commonly called feedforward, because their graph is a directed acyclic graph. Networks with cycles are commonly called recurrent. Such networks are commonly depicted in the manner shown at the top of the figure, where $\scriptstyle f$ is shown as being dependent upon itself. However, an implied temporal dependence is not shown.

### Learning

What has attracted the most interest in neural networks is the possibility of learning. Given a specific task to solve, and a class of functions $\scriptstyle F$, learning means using a set of observations to find $\scriptstyle f^{*} \in F$ which solves the task in some optimal sense.

This entails defining a cost function $\scriptstyle C : F \rightarrow \mathbb{R}$ such that, for the optimal solution $\scriptstyle f^*$, $\scriptstyle C(f^*) \leq C(f)$ $\scriptstyle \forall f \in F$ - i.e., no solution has a cost less than the cost of the optimal solution (see Mathematical optimization).

The cost function $\scriptstyle C$ is an important concept in learning, as it is a measure of how far away a particular solution is from an optimal solution to the problem to be solved. Learning algorithms search through the solution space to find a function that has the smallest possible cost.

For applications where the solution is dependent on some data, the cost must necessarily be a function of the observations, otherwise we would not be modelling anything related to the data. It is frequently defined as a statistic to which only approximations can be made. As a simple example, consider the problem of finding the model $\scriptstyle f$, which minimizes $\scriptstyle C=E\left[(f(x) - y)^2\right]$, for data pairs $\scriptstyle (x,y)$ drawn from some distribution $\scriptstyle \mathcal{D}$. In practical situations we would only have $\scriptstyle N$ samples from $\scriptstyle \mathcal{D}$ and thus, for the above example, we would only minimize $\scriptstyle \hat{C}=\frac{1}{N}\sum_{i=1}^N (f(x_i)-y_i)^2$. Thus, the cost is minimized over a sample of the data rather than the entire data set.

When $\scriptstyle N \rightarrow \infty$ some form of online machine learning must be used, where the cost is partially minimized as each new example is seen. While online machine learning is often used when $\scriptstyle \mathcal{D}$ is fixed, it is most useful in the case where the distribution changes slowly over time. In neural network methods, some form of online machine learning is frequently used for finite datasets.

See also: Mathematical optimization, Estimation theory, and Machine learning

#### Choosing a cost function

While it is possible to define some arbitrary, ad hoc cost function, frequently a particular cost will be used, either because it has desirable properties (such as convexity) or because it arises naturally from a particular formulation of the problem (e.g., in a probabilistic formulation the posterior probability of the model can be used as an inverse cost). Ultimately, the cost function will depend on the desired task. An overview of the three (3) main categories of learning tasks is provided below.

### Learning paradigms

There are three major learning paradigms, each corresponding to a particular abstract learning task. These are supervised learning, unsupervised learning and reinforcement learning.

#### Supervised learning

In supervised learning, we are given a set of example pairs $\scriptstyle (x, y), x \in X, y \in Y$ and the aim is to find a function $\scriptstyle f : X \rightarrow Y$ in the allowed class of functions that matches the examples. In other words, we wish to infer the mapping implied by the data; the cost function is related to the mismatch between our mapping and the data and it implicitly contains prior knowledge about the problem domain.

A commonly used cost is the mean-squared error, which tries to minimize the average squared error between the network's output, f(x), and the target value y over all the example pairs. When one tries to minimize this cost using gradient descent for the class of neural networks called multilayer perceptrons, one obtains the common and well-known backpropagation algorithm for training neural networks.

Tasks that fall within the paradigm of supervised learning are pattern recognition (also known as classification) and regression (also known as function approximation). The supervised learning paradigm is also applicable to sequential data (e.g., for speech and gesture recognition). This can be thought of as learning with a "teacher," in the form of a function that provides continuous feedback on the quality of solutions obtained thus far.

#### Unsupervised learning

In unsupervised learning, some data $\scriptstyle x$ is given and the cost function to be minimized, that can be any function of the data $\scriptstyle x$ and the network's output, $\scriptstyle f$.

The cost function is dependent on the task (what we are trying to model) and our a priori assumptions (the implicit properties of our model, its parameters and the observed variables).

As a trivial example, consider the model $\scriptstyle f(x) = a$, where $\scriptstyle a$ is a constant and the cost $\scriptstyle C=E[(x - f(x))^2]$. Minimizing this cost will give us a value of $\scriptstyle a$ that is equal to the mean of the data. The cost function can be much more complicated. Its form depends on the application: for example, in compression it could be related to the mutual information between $\scriptstyle x$ and $\scriptstyle f(x)$, whereas in statistical modeling, it could be related to the posterior probability of the model given the data. (Note that in both of those examples those quantities would be maximized rather than minimized).

Tasks that fall within the paradigm of unsupervised learning are in general estimation problems; the applications include clustering, the estimation of statistical distributions, compression and filtering.

#### Reinforcement learning

In reinforcement learning, data $\scriptstyle x$ are usually not given, but generated by an agent's interactions with the environment. At each point in time $\scriptstyle t$, the agent performs an action $\scriptstyle y_t$ and the environment generates an observation $\scriptstyle x_t$ and an instantaneous cost $\scriptstyle c_t$, according to some (usually unknown) dynamics. The aim is to discover a policy for selecting actions that minimizes some measure of a long-term cost; i.e., the expected cumulative cost. The environment's dynamics and the long-term cost for each policy are usually unknown, but can be estimated.

More formally, the environment is modeled as a Markov decision process (MDP) with states $\scriptstyle {s_1,...,s_n}\in S$ and actions $\scriptstyle {a_1,...,a_m} \in A$ with the following probability distributions: the instantaneous cost distribution $\scriptstyle P(c_t|s_t)$, the observation distribution $\scriptstyle P(x_t|s_t)$ and the transition $\scriptstyle P(s_{t+1}|s_t, a_t)$, while a policy is defined as conditional distribution over actions given the observations. Taken together, the two define a Markov chain (MC). The aim is to discover the policy that minimizes the cost; i.e., the MC for which the cost is minimal.

ANNs are frequently used in reinforcement learning as part of the overall algorithm. Dynamic programming has been coupled with ANNs (Neuro dynamic programming) by Bertsekas and Tsitsiklis[2] and applied to multi-dimensional nonlinear problems such as those involved in vehicle routing or natural resources management because of the ability of ANNs to mitigate losses of accuracy even when reducing the discretization grid density for numerically approximating the solution of the original control problems.

Tasks that fall within the paradigm of reinforcement learning are control problems, games and other sequential decision making tasks.

See also: dynamic programming and stochastic control

### Learning algorithms

Training a neural network model essentially means selecting one model from the set of allowed models (or, in a Bayesian framework, determining a distribution over the set of allowed models) that minimizes the cost criterion. There are numerous algorithms available for training neural network models; most of them can be viewed as a straightforward application of optimization theory and statistical estimation.

Most of the algorithms used in training artificial neural networks employ some form of gradient descent. This is done by simply taking the derivative of the cost function with respect to the network parameters and then changing those parameters in a gradient-related direction.

Evolutionary methods,[3] gene expression programming,[4] simulated annealing,[5] expectation-maximization, non-parametric methods and particle swarm optimization[6] are some commonly used methods for training neural networks.

See also: machine learning

## Employing artificial neural networks

Perhaps the greatest advantage of ANNs is their ability to be used as an arbitrary function approximation mechanism that 'learns' from observed data. However, using them is not so straightforward and a relatively good understanding of the underlying theory is essential.

• Choice of model: This will depend on the data representation and the application. Overly complex models tend to lead to problems with learning.
• Learning algorithm: There are numerous trade-offs between learning algorithms. Almost any algorithm will work well with the correct hyperparameters for training on a particular fixed data set. However selecting and tuning an algorithm for training on unseen data requires a significant amount of experimentation.
• Robustness: If the model, cost function and learning algorithm are selected appropriately the resulting ANN can be extremely robust.

With the correct implementation, ANNs can be used naturally in online learning and large data set applications. Their simple implementation and the existence of mostly local dependencies exhibited in the structure allows for fast, parallel implementations in hardware.

## Applications

The utility of artificial neural network models lies in the fact that they can be used to infer a function from observations. This is particularly useful in applications where the complexity of the data or task makes the design of such a function by hand impractical.

### Real-life applications

The tasks artificial neural networks are applied to tend to fall within the following broad categories:

• Function approximation, or regression analysis, including time series prediction, fitness approximation and modeling.
• Classification, including pattern and sequence recognition, novelty detection and sequential decision making.
• Data processing, including filtering, clustering, blind source separation and compression.
• Robotics, including directing manipulators, Computer numerical control.

Application areas include system identification and control (vehicle control, process control, natural resources management), quantum chemistry,[7] game-playing and decision making (backgammon, chess, poker), pattern recognition (radar systems, face identification, object recognition and more), sequence recognition (gesture, speech, handwritten text recognition), medical diagnosis, financial applications (automated trading systems), data mining (or knowledge discovery in databases, "KDD"), visualization and e-mail spam filtering.

Artificial neural networks have also been used to diagnose several cancers. An ANN based hybrid lung cancer detection system named HLND improves the accuracy of diagnosis and the speed of lung cancer radiology.[8] These networks have also been used to diagnose prostate cancer. The diagnoses can be used to make specific models taken from a large group of patients compared to information of one given patient. The models do not depend on assumptions about correlations of different variables. Colorectal cancer has also been predicted using the neural networks. Neural networks could predict the outcome for a patient with colorectal cancer with a lot more accuracy than the current clinical methods. After training, the networks could predict multiple patient outcomes from unrelated institutions.[9]

### Neural networks and neuroscience

Theoretical and computational neuroscience is the field concerned with the theoretical analysis and computational modeling of biological neural systems. Since neural systems are intimately related to cognitive processes and behavior, the field is closely related to cognitive and behavioral modeling.

The aim of the field is to create models of biological neural systems in order to understand how biological systems work. To gain this understanding, neuroscientists strive to make a link between observed biological processes (data), biologically plausible mechanisms for neural processing and learning (biological neural network models) and theory (statistical learning theory and information theory).

#### Types of models

Many models are used in the field defined at different levels of abstraction and modeling different aspects of neural systems. They range from models of the short-term behavior of individual neurons, models of how the dynamics of neural circuitry arise from interactions between individual neurons and finally to models of how behavior can arise from abstract neural modules that represent complete subsystems. These include models of the long-term, and short-term plasticity, of neural systems and their relations to learning and memory from the individual neuron to the system level.

## Neural network software

Main article: Neural network software

Neural network software is used to simulate, research, develop and apply artificial neural networks, biological neural networks and, in some cases, a wider array of adaptive systems.

## Types of artificial neural networks

Main article: Types of artificial neural networks

Artificial neural network types vary from those with only one or two layers of single direction logic, to complicated multi–input many directional feedback loops and layers. On the whole, these systems use algorithms in their programming to determine control and organization of their functions. Some may be as simple as a one-neuron layer with an input and an output, and others can mimic complex systems such as dANN, which can mimic chromosomal DNA through sizes at the cellular level, into artificial organisms and simulate reproduction, mutation and population sizes.[10]

Most systems use "weights" to change the parameters of the throughput and the varying connections to the neurons. Artificial neural networks can be autonomous and learn by input from outside "teachers" or even self-teaching from written-in rules.

## Theoretical properties

### Computational power

The multi-layer perceptron (MLP) is a universal function approximator, as proven by the Cybenko theorem. However, the proof is not constructive regarding the number of neurons required or the settings of the weights.

Work by Hava Siegelmann and Eduardo D. Sontag has provided a proof that a specific recurrent architecture with rational valued weights (as opposed to full precision real number-valued weights) has the full power of a Universal Turing Machine[11] using a finite number of neurons and standard linear connections. They have further shown that the use of irrational values for weights results in a machine with super-Turing power.[citation needed]

### Capacity

Artificial neural network models have a property called 'capacity', which roughly corresponds to their ability to model any given function. It is related to the amount of information that can be stored in the network and to the notion of complexity.

### Convergence

Nothing can be said in general about convergence since it depends on a number of factors. Firstly, there may exist many local minima. This depends on the cost function and the model. Secondly, the optimization method used might not be guaranteed to converge when far away from a local minimum. Thirdly, for a very large amount of data or parameters, some methods become impractical. In general, it has been found that theoretical guarantees regarding convergence are an unreliable guide to practical application.[citation needed]

### Generalization and statistics

In applications where the goal is to create a system that generalizes well in unseen examples, the problem of over-training has emerged. This arises in convoluted or over-specified systems when the capacity of the network significantly exceeds the needed free parameters. There are two schools of thought for avoiding this problem: The first is to use cross-validation and similar techniques to check for the presence of overtraining and optimally select hyperparameters such as to minimize the generalization error. The second is to use some form of regularization[disambiguation needed]. This is a concept that emerges naturally in a probabilistic (Bayesian) framework, where the regularization can be performed by selecting a larger prior probability over simpler models; but also in statistical learning theory, where the goal is to minimize over two quantities: the 'empirical risk' and the 'structural risk', which roughly corresponds to the error over the training set and the predicted error in unseen data due to overfitting.

Confidence analysis of a neural network

Supervised neural networks that use an MSE cost function can use formal statistical methods to determine the confidence of the trained model. The MSE on a validation set can be used as an estimate for variance. This value can then be used to calculate the confidence interval of the output of the network, assuming a normal distribution. A confidence analysis made this way is statistically valid as long as the output probability distribution stays the same and the network is not modified.

By assigning a softmax activation function on the output layer of the neural network (or a softmax component in a component-based neural network) for categorical target variables, the outputs can be interpreted as posterior probabilities. This is very useful in classification as it gives a certainty measure on classifications.

The softmax activation function is:

$y_i=\frac{e^{x_i}}{\sum_{j=1}^c e^{x_j}}$

### Dynamic properties

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Various techniques originally developed for studying disordered magnetic systems (i.e., the spin glass) have been successfully applied to simple neural network architectures, such as the Hopfield network. Influential work by E. Gardner and B. Derrida has revealed many interesting properties about perceptrons with real-valued synaptic weights, while later work by W. Krauth and M. Mezard has extended these principles to binary-valued synapses.

## Disadvantages

One drawback to using artificial neural networks, particularly in robotics, is that they require a large diversity of training for real-world operation. A. K. Dewdney, a former Scientific American columnist, wrote in 1997, "Although neural nets do solve a few toy problems, their powers of computation are so limited that I am surprised anyone takes them seriously as a general problem-solving tool." (Dewdney, p. 82)

Arguments for Dewdney's position are that to implement large and effective software neural networks, much processing and storage resources need to be committed. While the brain has hardware tailored to the task of processing signals through a graph of neurons, simulating even a most simplified form on Von Neumann technology may compel a NN designer to fill many millions of database rows for its connections - which can lead to excessive RAM and HD necessities. Furthermore, the designer of NN systems will often need to simulate the transmission of signals through many of these connections and their associated neurons - which must often be matched with incredible amounts of CPU processing power and time. While neural networks often yield effective programs, they too often do so at the cost of time and monetary efficiency.

Arguments against Dewdney's position are that neural nets have been successfully used to solve many complex and diverse tasks, ranging from autonomously flying aircraft[12] to detecting credit card fraud.[13] Technology writer Roger Bridgman commented on Dewdney's statements about neural nets:

Neural networks, for instance, are in the dock not only because they have been hyped to high heaven, (what hasn't?) but also because you could create a successful net without understanding how it worked: the bunch of numbers that captures its behaviour would in all probability be "an opaque, unreadable table...valueless as a scientific resource". In spite of his emphatic declaration that science is not technology, Dewdney seems here to pillory neural nets as bad science when most of those devising them are just trying to be good engineers. An unreadable table that a useful machine could read would still be well worth having.[14]

Some other criticisms came from believers of hybrid models (combining neural networks and symbolic approaches). They advocate the intermix of these two approaches and believe that hybrid models can better capture the mechanisms of the human mind (Sun and Bookman 1994).

## Successes in Pattern Recognition Contests since 2009

Between 2009 and 2012, the recurrent neural networks and deep feedforward neural networks developed in the research group of Jürgen Schmidhuber at the Swiss AI Lab IDSIA have won eight international competitions in pattern recognition and machine learning.[15] For example, the bi-directional and multi-dimensional Long short term memory (LSTM)[16][17] of Alex Graves et al. won three competitions in connected handwriting recognition at the 2009 International Conference on Document Analysis and Recognition (ICDAR), without any prior knowledge about the three different languages to be learned. Fast GPU-based implementations of this approach by Dan Ciresan and colleagues at IDSIA have won several pattern recognition contests, including the IJCNN 2011 Traffic Sign Recognition Competition,[18] the ISBI 2012 Segmentation of Neuronal Structures in Electron Microscopy Stacks challenge,[19] and others. Their neural networks also were the first artificial pattern recognizers to achieve human-competitive or even superhuman performance[20] on important benchmarks such as traffic sign recognition (IJCNN 2012), or the famous MNIST handwritten digits problem of Yann LeCun at NYU. Deep, highly nonlinear neural architectures similar to the 1980 Neocognitron by Kunihiko Fukushima[21] and the "standard architecture of vision"[22] can also be pre-trained by unsupervised methods[23][24] of Geoff Hinton's lab at Toronto University. A team from this lab won a 2012 contest sponsored by Merck to design software to help find molecules that might lead to new drugs.[25]

## Gallery

• A single-layer feedforward artificial neural network. Arrows originating from $\scriptstyle x_2$ are omitted for clarity. There are p inputs to this network and q outputs. There is no activation function (or equivalently, the activation function is $\scriptstyle g(x)=x$). In this system, the value of the qth output, $\scriptstyle y_q$ would be calculated as $\scriptstyle y_q = \sum(x_i*w_{iq})$

• A two-layer feedforward artificial neural network.

## References

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2. Bertsekas, D.P., Tsitsiklis, J.N. (1996). Neuro-dynamic programming. Athena Scientific. p. 512. ISBN 1-886529-10-8.
3. de Rigo, D., Castelletti, A., Rizzoli, A.E., Soncini-Sessa, R., Weber, E. (January 2005). "A selective improvement technique for fastening Neuro-Dynamic Programming in Water Resources Network Management". In Pavel Zítek. Proceedings of the 16th IFAC World Congress - IFAC-PapersOnLine. 16. 16th IFAC World Congress. Prague, Czech Republic: IFAC. doi:10.3182/20050703-6-CZ-1902.02172. ISBN 978-3-902661-75-3. Retrieved 2011-12-30.
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5. Da, Y., Xiurun, G. (July 2005). "An improved PSO-based ANN with simulated annealing technique". In T. Villmann. New Aspects in Neurocomputing: 11th European Symposium on Artificial Neural Networks. Elsevier. doi:10.1016/j.neucom.2004.07.002. Retrieved 2011-12-30.
6. Wu, J., Chen, E. (May 2009). "A Novel Nonparametric Regression Ensemble for Rainfall Forecasting Using Particle Swarm Optimization Technique Coupled with Artificial Neural Network". In Wang, H., Shen, Y., Huang, T., Zeng, Z.. 6th International Symposium on Neural Networks, ISNN 2009. Springer. doi:10.1007/978-3-642-01513-7_6. ISBN 978-3-642-01215-0. Retrieved 2012-01-01.
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12. "NASA NEURAL NETWORK PROJECT PASSES MILESTONE". NASA. Retrieved 12 July 2010.
13. "Counterfeit Fraud" (PDF). VISA. p. 1. Retrieved 12 July 2010. "Neural Networks (24/7 Monitoring):"
14. 2012 Kurzweil AI Interview with Jürgen Schmidhuber on the eight competitions won by his Deep Learning team 2009-2012
15. Graves, Alex; and Schmidhuber, Jürgen; Offline Handwriting Recognition with Multidimensional Recurrent Neural Networks, in Bengio, Yoshua; Schuurmans, Dale; Lafferty, John; Williams, Chris K. I.; and Culotta, Aron (eds.), Advances in Neural Information Processing Systems 22 (NIPS'22), December 7th–10th, 2009, Vancouver, BC, Neural Information Processing Systems (NIPS) Foundation, 2009, pp. 545–552
16. D. Ciresan, A. Giusti, L. Gambardella, J. Schmidhuber. Deep Neural Networks Segment Neuronal Membranes in Electron Microscopy Images. In Advances in Neural Information Processing Systems (NIPS 2012), Lake Tahoe, 2012.
17. K. Fukushima. Neocognitron: A self-organizing neural network model for a mechanism of pattern recognition unaffected by shift in position. Biological Cybernetics, 36(4): 93-202, 1980.
18. [[Geoff Hinton|Hinton, G. E.]]; Osindero, S.; Teh, Y. (2006). "A fast learning algorithm for deep belief nets". 18 (7): 1527–1554. doi:10.1162/neco.2006.18.7.1527. PMID 16764513.

28. Boddhu, Sanjay K., John C. Gallagher, and Saranyan A. Vigraham. "A commercial off-the-shelf implementation of an analog neural computer." International Journal on Artificial Intelligence Tools 17.02 (2008): 241-258.

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