Neural networks

A neural network is a type of computational architecture that is composed of layers of perceptrons. A perceptron, first conceived  in the 1950s by Frank Rosenblatt, models the biological neuron and computes a linear combination of a vector of input. It also outputs a transformation of the linear combination using a non-linear activation, such as the sigmoid function. Suppose a perceptron receives an input vector of . The output, a, of the perceptron, would be as follows:

Whereare the weights of the perceptron, b is a constant, called the bias, andis the sigmoid activation function that outputs a value between 0 and 1.

Perceptrons have been widely used as a computational model to make decisions. Suppose the task was to predict the likelihood of sunny weather the next day. Eachwould represent a variable, such as the temperature of the current day, humidity, or the weather of the previous day. Then,would compute a value that reflects how likely it is that there will be sunny weather tomorrow. If the model has a good set of values for , it is able to make accurate decisions.

In a typical neural network, there are multiple layers of neurons, where each neuron in a given layer is connected to all neurons in the prior and subsequent layers. Hence these layers are also referred to as fully-connected layers. The weights of a given layer, l, can be represented as a matrix, W^l:

Where each w_ij denotes the weight between the i neuron of the previous layer and the j neuron of this layer. B^l denotes a vector of biases, one for each neuron in the l layer. Hence, the activation, a^l, of a given layer, l, can be defined as follows:

Where a⁰(x) is just the input. Such neural networks with multiple layers of neurons are called multilayer perceptrons (MLP). There are three components in an MLP: the input layer, the hidden layers, and the output layer. The data flows from the input layer, transformed through a series of linear and non-linear functions in the hidden layers, and is outputted from the output layer as a decision or a prediction. Hence this architecture is also referred to as a feed-forward network. The following diagram shows what a fully-connected network would look like: