In this chapter, we have discussed Hebb's rule, showing how it can drive the computation of the first principal component of the input dataset. We have also seen that this rule is unstable because it leads to the infinite growth of the synaptic weights and how it's possible to solve this problem using normalization or Oja's rule. 

We have introduced two different neural networks based on Hebbian learning (Sanger's and Rubner-Tavan's networks), whose internal dynamics are slightly different, which are able to extract the first n principal components in the right order (starting from the largest eigenvalue) without eigendecomposing the input covariance matrix.

Finally, we have introduced the concept of SOM and presented a model called a Kohonen network, which is able to map the input patterns onto a surface where some attractors (one per class) are placed through a competitive learning process. Such a model is able to recognize new patterns (belonging to the same distribution) by eliciting a strong response in the attractor, that is most similar to the pattern. In this way, after a labeling process, the model can be employed as a soft classifier that can easily manage noisy or altered patterns.

In the next chapter, we're going to discuss some important clustering algorithms, focusing on the difference (already discussed in the previous chapters) between hard and soft clustering and discussing the main techniques employed to evaluate the performance of an algorithm.