Manifold learning

In Chapter 02, Introduction to Semi-Supervised Learning, we discussed the manifold assumption, saying that high-dimensional data normally lies on low-dimensional manifolds. Of course, this is not a theorem, but in many real cases, the assumption is proven to be correct, and it allows us to work with non-linear dimensionality reduction algorithms that would be otherwise unacceptable. In this section, we're going to analyze some of these algorithms. They are all implemented in Scikit-Learn, therefore it's easy to try them with complex datasets.

Isomap

Isomap is one of the simplest algorithms, and it's based on the idea of reducing the dimensionality while trying to preserve the geodesic distances measured on the original manifold where the input data lies. The algorithm works in three steps. The first operation is a k-nearest neighbors clustering and the construction of the following graph. The vertices will be the samples, while the edges represent the connections among nearest...