Editing Perceptron (section)

== Definition ==
[[Image:Perceptron.svg|right|thumb|353x353px|The appropriate weights are applied to the inputs, and the resulting weighted sum passed to a function that produces the output o.]]In the modern sense, the perceptron is an algorithm for learning a binary classifier called a [[Linear classifier#Definition|threshold function]]: a function that maps its input <math>\mathbf{x}</math> (a real-valued [[Vector space|vector]]) to an output value <math>f(\mathbf{x})</math> (a single [[Binary function|binary]] value):

<math display="block">
f(\mathbf{x}) = h(\mathbf{w} \cdot \mathbf{x} + b)
</math>

where <math>h</math> is the [[Heaviside step function|Heaviside step-function]] (where an input of <math display="inline"> > 0</math> outputs 1; otherwise 0 is the output ), <math>\mathbf{w}</math> is a vector of real-valued weights, <math>\mathbf{w} \cdot \mathbf{x}</math> is the [[dot product]] <math display="inline">\sum_{i=1}^m w_i x_i</math>, where {{mvar|m}} is the number of inputs to the perceptron, and {{mvar|b}} is the ''bias''. The bias shifts the decision boundary away from the origin and does not depend on any input value.

Equivalently, since <math>\mathbf{w}\cdot \mathbf{x} + b = (\mathbf{w}, b) \cdot (\mathbf{x}, 1)</math>, we can add the bias term <math>b</math> as another weight <math>\mathbf{w}_{m+1}</math> and add a coordinate <math>1</math> to each input <math>\mathbf{x}</math>, and then write it as a linear classifier that passes the origin:<math display="block">
f(\mathbf{x}) = h(\mathbf{w} \cdot \mathbf{x})
</math>

The binary value of <math>f(\mathbf{x})</math> (0 or 1) is used to perform binary classification on <math>\mathbf{x}</math> as either a positive or a negative instance. Spatially, the bias shifts the position (though not the orientation) of the planar [[decision boundary]].

In the context of neural networks, a perceptron is an [[artificial neuron]] using the [[Heaviside step function]] as the activation function. The perceptron algorithm is also termed the '''single-layer perceptron''', to distinguish it from a [[multilayer perceptron]], which is a misnomer for a more complicated neural network.  As a linear classifier, the single-layer perceptron is the simplest [[feedforward neural network]].