# Semi-supervised Support Vector Machines (S^{3}VM)

When we discussed the cluster assumption in the previous chapter, we also defined the low-density regions as boundaries and the corresponding problem as low-density separation. A common supervised classifier based on this concept is a **Support Vector Machine** (**SVM**), the objective of which is to maximize the distance between the dense regions where the samples must be.

## S^{3}VM Theory

For a complete description of linear and kernel-based SVMs, please refer to Bonaccorso G., *Machine Learning Algorithms, Second Edition*, Packt Publishing, 2018. However, it's useful to remind yourself of the basic model for a linear SVM with slack variables :

This model is based on the assumption that *y*_{i} can be either -1 or 1. The slack variables or soft-margins are variables, one for each sample, introduced to reduce the *strength* imposed by the original condition (*min ||w||*), which is based on a hard margin that misclassifies...