Single-Layer Neural networks: Perceptron

If we consider a single neuron without an activation function, the analysis of it can be formulated in terms of signal processing, viewing the neuron as a linear system. Refer to the adaptative filtering of the signal processing reference for more insight on this.

With a Hard limitter activation function, the perceptron can be view as two decision regions separated by the hyperplane.

$$\sum_{i=1}^{m} w_i x_i + b = 0 = \ve{w}^t \ve{x}$$ (1)

$$y(n) = \varphi((1,w_1(n),\hdots,w_m(n)) \begin{pmatrix} b(n... ...vdots\\ x_m(n) \end{pmatrix}) = \varphi(\ve{w}_n^t \ve{x}_n)$$ (2)

In this case $\varphi(\cdot) = sgn(\cdot)$. When $y=+1$ pattern belongs to class 0 $C_0$, when $y=-1$, pattern belongs to class 1 $C_1$.

• The perceptron can classify the input vectors correctly if and only if the input vectors are linearly separable.
• A weight vector that does the classification correctly is found after a finite number of iterations