AI News, Artificial Neural Networks/Print Version

Artificial Neural Networks/Print Version

This book is going to be aimed at advanced undergraduates and graduate students in the areas of computer science, mathematics, engineering, and the sciences.

Artificial Neural Networks, also known as “Artificial neural nets”, “neural nets”, or ANN for short, are a computational tool modeled on the interconnection of the neuron in the nervous systems of the human brain and that of other organisms.

neural nets are a type of non-linear processing system that is ideally suited for a wide range of tasks, especially tasks where there is no existing algorithm for task completion.

With proper training, ANN are capable of generalization, the ability to recognize similarities among different input patterns, especially patterns that have been corrupted by noise.

Each neuron is a multiple-input, multiple-output (MIMO) system that receives signals from the inputs, produces a resultant signal, and transmits that signal to all outputs.

However, to reproduce the effect of the synapse, the connections between PE are assigned multiplicative weights, which can be calibrated or “trained” to produce the proper system output.

Where ζ is the weighted sum of the inputs (the inner product of the input vector and the tap-weight vector), and σ(ζ) is a function of the weighted sum.

If we recognize that the weight and input elements form vectors w and x, the ζ weighted sum becomes a simple dot product:

The dotted line in the center of the neuron represents the division between the calculation of the input sum using the weight vector, and the calculation of the output value using the activation function.

Neural networks tend to have one input per degree of freedom in the input space, and one output per degree of freedom in the output space.

Expert systems, by contrast, are used in situations where there is insufficient data and theoretical background to create any kind of a reliable problem model.

Expert systems emulate the deduction processes of a human expert, by collecting information and traversing the solution space in a directed manner.

Though such assumptions are not required, it has been found that the addition of such a priori information as the statistical distribution of the input space can help to speed training.

During training, the neural network performs the necessary analytical work, which would require non-trivial effort on the part of the analyst if other methods were to be used.

During training, care must be taken not to provide too many input examples and different numbers of training examples could produce very different results in the quality and robustness of the network.

Some of the more important parameters in terms of training and network capacity are the number of hidden neurons, the learning rate and the momentum parameter.

These neurons are essentially hidden from view, and their number and organization can typically be treated as a black box to people who are interfacing with the system.

Square root of the sum of squared differences between the network targets and actual outputs divided by number of patterns (only for training by minimum error).

In the case of a biological neural net, neurons are living cells with axons and dendrites that form interconnections through electro-chemical synapses.

These neurotransmitters, along with other chemicals present in the synapse form the message that is received by the post-synaptic membrane of the dendrite of the next cell, which in turn is converted to an electrical signal.

This page is going to provide a brief overview of biological neural networks, but the reader will have to find a better source for a more in-depth coverage of the subject.

Neurons utilize a threshold mechanism, so that signals below a certain threshold are ignored, but signals above the threshold cause the neuron to fire.

The random interconnection at the cellular level is rendered into a computational tool by the learning process of the synapse, and the formation of new synapses between nearby neurons.

The history of neural networking arguably started in the late 1800s with scientific attempts to study the workings of the human brain.

The Mark I was a two layer Perceptron, Hecht-Nielsen showed in 1990 that a three layer machine (multi layer Perceptron, or MLP) was capable of solving nonlinear separation problems.

Even though this book is going to focus on MATLAB for its problems and examples, there are a number of other tools that can be used for constructing, testing, and implementing neural networks.

The output is a certain value, A1, if the input sum is above a certain threshold and A0 if the input sum is below a certain threshold.

linear combination is where the weighted sum input of the neuron plus a linearly dependent bias becomes the system output.

In these cases, the sign of the output is considered to be equivalent to the 1 or 0 of the step function systems, which enables the two methods be to equivalent if

This is called the log-sigmoid because a sigmoid can also be constructed using the hyperbolic tangent function instead of this relation, in which case it would be called a tan-sigmoid.

1

+

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−

β

t

If ρl is a vector of activation functions [σ1 σ2 … σn] that acts on each row of input and bl is an arbitrary offset vector (for generalization) then the total output of layer l is given as:

hidden layer neuron represents a basis function of the output space, with respect to a particular center in the input space.

In a recurrent network, the weight matrix for each layer l contains input weights from all other neurons in the network, not just neurons from the previous layer.

context layer feeds the hidden layer at iteration N with a value computed from the output of the hidden layer at iteration N-1, providing a short memory effect.

Because the only tap weights modified during training are the output layer tap weights, training is typically quick and computationally efficient in comparison to other multi-layer networks that are not sparsely connected.

This means that mathematical minimization or optimization problems can be solved automatically by the Hopfield network if that problem can be formulated in terms of the network energy.

Each attractor represents a different data value that is stored in the network, and a range of associated patterns can be used to retrieve the data pattern.

SOM are modeled on biological neural networks, where groups of neurons appear to self organize into specific regions with common functionality.

The Euclidean distance from each input sample to the weight vector of each neuron is computed, and the neuron whose weight vector is most similar to the input is declared the best match unit (BMU).

In adaptive resonance theory (ART) networks, an overabundance of neurons leads some neurons to be committed (active) and others to be uncommitted (inactive).

Where: Here, pij is the probability that elements i and j will both be on when the system is in its training phase (positive phase), and qij is the probability that both elements i and j will be on during the production phase (negative phase).

Because Boltzmann machine weight updates only require looking at the expected distributions of surrounding neurons, it is a plausible model for how actual biological neural networks learn.

For instance, the most common answer among a discrete set of answers in the committee can be taken as the overall answer, or the average answer can be taken.

Committee of machines (COM) systems tend to be more robust then the individual component systems, but they can also lose some of the “expertise” of the individual systems when answers are averaged out.

Given an input set x, and a cost function g(x, y) of the input and output sets, the goal is to minimize the cost function through a proper selection of f (the relationship between x, and y).

Unsupervised learning is useful in situations where a cost function is known, but a data set is not know that minimizes that cost function over a particular input space.

Error-Correction Learning, used with supervised learning, is the technique of comparing the system output to the desired output value, and using that error to direct the training.

In the most direct route, the error values can be used to directly adjust the tap weights, using an algorithm such as the backpropagation algorithm.

By following the path of steepest descent at each iteration, we will either find a minimum, or the algorithm could diverge if the weight space is infinitely decreasing.

The backpropagation algorithm, in combination with a supervised error-correction learning rule, is one of the most popular and robust tools in the training of artificial neural networks.

When talking about backpropagation, it is useful to define the term interlayer to be a layer of neurons, and the corresponding input tap weights to that layer.

Where xil-1 are the outputs from the previous interlayer (the inputs to the current interlayer), wijl is the tap weight from the i input from the previous interlayer to the j element of the current interlayer.

The backpropagation algorithm specifies that the tap weights of the network are updated iteratively during training to approach the minimum of the error function.

This makes it a plausible theory for biological learning methods, and also makes Hebbian learning processes ideal in VLSI hardware implementations where local signals are easier to obtain.

Neurons become trained to be individual feature detectors, and a combination of feature detectors can be used to identify large classes of features from the input space.

Adaptive Resonance Theory (ART) learning algorithms compare the weight vector, known as the prototype, to the current input vector to produce a distance, r.

When a new input sequence is detected that does not resonate with any committed nodes, an uncommitted node is committed, and it’s prototype vector is set to the current input vector.

SOM are modeled on biological neural networks, where groups of neurons appear to self organize into specific regions with common functionality.

The Euclidean distance from each input sample to the weight vector of each neuron is computed, and the neuron whose weight vector is most similar to the input is declared the best match unit (BMU).

Similar to pattern matching, clustering is the ability to associate similar input patterns together, based on a measurement of their similarity or dissimilarity.

Feature detection or “association” networks are trained using non-noisy data, in order to recognize similar patterns in noisy or incomplete data.

ANN can be trained to match the statistical properties of a particular input signal, and can even be used to predict future values of time series.

Meteorological prediction is a difficult process because current atmospheric models rely on highly recursive sets of differential equations which can be difficult to calculate, and which propagate errors through the successive iterations.

Once the relation has been modeled to the necessary accuracy by the network, it can be used for a variety of tasks, such as series prediction, function approximation, and function optimization.

Function approximation or modeling is the act of training a neural network using a given set of input-output data (typically through supervised learning) in order to deduce the relationship between the input and the output.

Because of the modular and non-linear nature of artificial neural nets, they are considered to be able to approximate any arbitrary function to an arbitrary degree of accuracy.

Artificial neural networks have been employed for use in control systems because of their ability to identify patterns, and to match arbitrary non-linear response curves.

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