Mastering Machine Learning Algorithms

Manifold assumption

This is the less intuitive assumption, but it can be extremely useful to reduce the complexity of many problems. First of all, we can provide a non-rigorous definition of a manifold. An n-manifold is a topological space that is globally curved, but locally homeomorphic to an n-dimensional Euclidean space. In the following diagram, there's an example of a manifold: the surface of a sphere in ℜ³:

2D manifold obtained from a spherical surface

The small patch around P (for ε → 0) can be mapped to a flat circular surface. Therefore, the properties of a manifold are locally based on the Euclidean geometry, while, globally, they ...