Perceptrons

A perceptron takes a vector of real-valued inputs, calculates a linear combination of these, and outputs a 1 if its greater than some threshold or -1 otherwise. Formally, $o (x_{1}, \dots, x_{n}) = {\begin{matrix} 1 & if w_{0} + w_{1} x_{1} + \cdot \cdot \cdot + w_{n} x_{n} > 0 \\ - 1 & otherwise. \end{matrix}$
Sometimes we'll use simpler vector notation $o (\overset{⇀}{x}) = {\begin{matrix} 1 & if \overset{⇀}{w} \cdot \overset{⇀}{x} > 0 \\ - 1 & otherwise. \end{matrix}$ where we assume an additional constant input $x_{0} = 1$ .
Learning involves choosing values for $w_{0}, \dots, w_{n}$ such that the output $o$ value matches the target value $t$ .