1 topological data analysis (TDA)

1.1 Betti numbers

• b_0 - number of components of the space
• b_1 - number of essentially different loops in the space
• b_2 - number of 3D-voids in the space

1.1.1 examples

1. Closed disk b_0 = 1, b_1 = 0, b_2 = 0
2. Two disjoint closed disks b_0 = 2, b_1 = 0, b_2 = 0
3. The circle b_0 = 1, b_1 = 1, b_2 = 0
4. Sphere b_0 = 1, b_1 = 0, b_2 = 1
5. Torus b_0 = 1, b_1 = 2, b_2 = 1

1.2 persistent homology

By introducing a proximity parameter (say r > 0), to any data set (finite metric space) we may assign an evolving family of spaces. To these we can find topological descriptors - Betti numbers.

1.3 topological entropy

Let P_i = (b_i, d_i) for i = 1, \cdots, n.

Let p_i = \frac{d_i - b_i}{\sum_{i=1}^n (d_i - b_i)}

Then entropy is defined as E = - \sum_{i=1}^n p_i \ln p_i

The vector of entropies forms a topological descriptor of the point cloud.

1.4 the mapper algorithm

1. divide the dataset into overlapping parts
2. to each part apply a clustering algorithm
3. if clusters from different parts have sufficient overlap, connect the vertices by an edge

1.5 homeomorphism (equivalence)

Two spaces A and B are said to be equivalent (homeomorphic) if there is a continuous map of spaces f: A \to B with a continuous inverse map f^{-1}: B \to A, then f is called the homeomorphism.