Artificial Neural Networks (ANN)

Table of Contents

1. Recall Perceptron


XOR Problem

  • Minsky-Papert Controversy on XOR
    • not linearly separable
    • limitation of perceptron

2. From Perceptron to Multi-Layer Perceptron (MLP)

2.1. Perceptron for $h_{\omega}(x)$

  • Neurons compute the weighted sum of their inputs

  • A neuron is activated or fired when the sum $a$ is positive

$$ \begin{align*} a &= \omega_0 + \omega_1 x_1 + \omega_2 x_2 \\ \\ \hat{y} &= g(a) = \begin{cases} 1 & a > 0\\ 0 & \text{otherwise} \end{cases} \end{align*} $$

  • A step function is not differentiable

  • One layer is often not enough
    • One hyperplane

2.2. Multi-layer Perceptron = Artificial Neural Networks (ANN)


Differentiable activation function

In a compact representation

Multi-layer perceptron

2.3. Another Perspective: ANN as Kernel Learning

We can represent this “neuron” as follows:

  • The main weakness of linear predictors is their lack of capacity. For classification, the populations have to be linearly separable.

  • The XOR example can be solved by pre-processing the data to make the two populations linearly separable.