# Geometry of binary threshold neurons and their networks

It performs a division of the space of inputs by a hyperplane. It is specially useful in the last layer of a network intended to perform binary classification of the inputs. It can be approximated from other sigmoidal functions by assigning large values to the weights. In this case, the output unit is simply the weighted sum of its inputs plus a bias term. A number of such linear neurons perform a linear transformation of the input vector.

This is usually more useful in the first layers of a network. A number of analysis tools exist based on linear models, such as harmonic analysis , and they can all be used in neural networks with this linear neuron. The bias term allows us to make affine transformations to the data. Linear transformation , Harmonic analysis , Linear filter , Wavelet , Principal component analysis , Independent component analysis , Deconvolution. A fairly simple non-linear function, the sigmoid function such as the logistic function also has an easily calculated derivative, which can be important when calculating the weight updates in the network.

It thus makes the network more easily manipulable mathematically, and was attractive to early computer scientists who needed to minimize the computational load of their simulations. It was previously commonly seen in multilayer perceptrons. However, recent work has shown sigmoid neurons to be less effective than rectified linear neurons.

The reason is that the gradients computed by the backpropagation algorithm tend to diminish towards zero as activations propagate through layers of sigmoidal neurons, making it difficult to optimize neural networks using multiple layers of sigmoidal neurons. The following is a simple pseudocode implementation of a single TLU which takes boolean inputs true or false , and returns a single boolean output when activated.

An object-oriented model is used. No method of training is defined, since several exist. If a purely functional model were used, the class TLU below would be replaced with a function TLU with input parameters threshold, weights, and inputs that returned a boolean value. From Wikipedia, the free encyclopedia. This section needs expansion.

You can help by adding to it. It has been suggested that this section be split out into another article titled transfer function. It has been suggested that this section be split out into another article titled Threshold Logic Unit. Weakly connected neural networks. Neural network models of birdsong production, learning, and coding PDF. The units in Hopfield nets are binary threshold units, i. Hopfield nets normally have units that take on values of 1 or -1, and this convention will be used throughout this page.

However, other literature might use units that take values of 0 and 1. The constraint that weights are symmetric guarantees that the energy function decreases monotonically while following the activation rules. Updating one unit node in the graph simulating the artificial neuron in the Hopfield network is performed using the following rule:. The weight between two units has a powerful impact upon the values of the neurons.

Thus, the values of neurons i and j will converge if the weight between them is positive. Similarly, they will diverge if the weight is negative. Hopfield nets have a scalar value associated with each state of the network referred to as the "energy", E, of the network, where:. This value is called the "energy" because: Furthermore, under repeated updating the network will eventually converge to a state which is a local minimum in the energy function which is considered to be a Lyapunov function.

Thus, if a state is a local minimum in the energy function, it is a stable state for the network. Note that this energy function belongs to a general class of models in physics , under the name of Ising models ; these in turn are a special case of Markov networks , since the associated probability measure , the Gibbs measure , has the Markov property. Initialization of the Hopfield Networks is done by setting the values of the units to the desired start pattern. Repeated updates are then performed until the network converges to an attractor pattern.

Convergence is generally assured, as Hopfield proved that the attractors of this nonlinear dynamical system are stable, not periodic or chaotic as in some other systems. Therefore, in the context of Hopfield Networks, an attractor pattern is a final stable state, a pattern that cannot change any value within it under updating.

Training a Hopfield net involves lowering the energy of states that the net should "remember". This allows the net to serve as a content addressable memory system, that is to say, the network will converge to a "remembered" state if it is given only part of the state. The net can be used to recover from a distorted input to the trained state that is most similar to that input. This is called associative memory because it recovers memories on the basis of similarity.

For example, if we train a Hopfield net with five units so that the state 1, -1, 1, -1, 1 is an energy minimum, and we give the network the state 1, -1, -1, -1, 1 it will converge to 1, -1, 1, -1, 1. Thus, the network is properly trained when the energy of states which the network should remember are local minima. There are various different learning rules that can be used to store information in the memory of the Hopfield Network.

It is desirable for a learning rule to have both of the following two properties:. These properties are desirable, since a learning rule satisfying them is more biologically plausible. For example, since the human brain is always learning new concepts, one can reason that human learning is incremental.

A learning system that were not incremental would generally be trained only once, with a huge batch of training data. This pulsing can be translated into continuous values. The rate activations per second, etc.

The faster a biological neuron fires, the faster nearby neurons accumulate electrical potential or lose electrical potential, depending on the "weighting" of the dendrite that connects to the neuron that fired. Research has shown that unary coding is used in the neural circuits responsible for birdsong production. Another contributing factor could be that unary coding provides a certain degree of error correction.

The model was specifically targeted as a computational model of the "nerve net" in the brain. Initially, only a simple model was considered, with binary inputs and outputs, some restrictions on the possible weights, and a more flexible threshold value.

Since the beginning it was already noticed that any boolean function could be implemented by networks of such devices, what is easily seen from the fact that one can implement the AND and OR functions, and use them in the disjunctive or the conjunctive normal form.

Researchers also soon realized that cyclic networks, with feedbacks through neurons, could define dynamical systems with memory, but most of the research concentrated and still does on strictly feed-forward networks because of the smaller difficulty they present. One important and pioneering artificial neural network that used the linear threshold function was the perceptron , developed by Frank Rosenblatt. This model already considered more flexible weight values in the neurons, and was used in machines with adaptive capabilities.

In the late s, when research on neural networks regained strength, neurons with more continuous shapes started to be considered. The possibility of differentiating the activation function allows the direct use of the gradient descent and other optimization algorithms for the adjustment of the weights. Neural networks also started to be used as a general function approximation model.

The best known training algorithm called backpropagation has been rediscovered several times but its first development goes back to the work of Paul Werbos. The transfer function of a neuron is chosen to have a number of properties which either enhance or simplify the network containing the neuron. Crucially, for instance, any multilayer perceptron using a linear transfer function has an equivalent single-layer network [ citation needed ] ; a non-linear function is therefore necessary to gain the advantages of a multi-layer network.

Below, u refers in all cases to the weighted sum of all the inputs to the neuron, i. The "signal" is sent, i. This function is used in perceptrons and often shows up in many other models.

It performs a division of the space of inputs by a hyperplane. It is specially useful in the last layer of a network intended to perform binary classification of the inputs. It can be approximated from other sigmoidal functions by assigning large values to the weights. In this case, the output unit is simply the weighted sum of its inputs plus a bias term.

A number of such linear neurons perform a linear transformation of the input vector.