Artificial Neural Networks (ANN)
Table of Contents
1. Recall Perceptron¶
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)¶
Multi-neurons
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.