6 Persistent Homology
“Given a growing simplicial complex, persistent homology encodes the evolution of its holes” – [Unknown]
We are now ready to fiscuss persistent homology. For more on these topics, you can refer to any of the following textbooks on applied topology/persistent homology, which will meet our immediate needs for this course.
- Introduction to Persistent Homology, Ziga Virk
- Computational Topology for Data Analysis, Tamal Dey and Yusu Wang
Those who want to learn more about homology can refer to the following book:
A filtration of simplicial complex K is a nested sequence of sub-complexes K_1 \subset K_2 \subset \cdots \subset K_N=K
and (for 1 \le n \le m \le N) inclusions i_{n,m}:K_n \hookrightarrow K_m. These inclusions induce maps (i_{n,m})^d_* on the homology groups in each dimension d.
H_d(K_1) \xrightarrow{(i_{1,2})^d_*} H_d(K_2) \xrightarrow{} \cdots \xrightarrow{} H_d(K_N)=H_d(K)
Exercise 6.1
- Explain how the Rips, \check{C}ech and Alpha complexes define filtrations.
- Describe the sublevel set filtration given by …
Let K be a simplicial complex and d \in \{0,1, \cdots\}. Given a filtration K_1 \xhookrightarrow{} K_2 \xhookrightarrow{} \cdots \xhookrightarrow{} K_N=K
the d-dimensional persistent homology groups with \mathbb{Z}_2-coefficients are the images of (i_{n,m})^d_*: H_d(K_n) \to H_d(K_m)
The corresponding ranks are the persistent Betti numbers
\beta^d_{n,m} =\textbf{rank} (i_{n,m})^d_*
More specifically:
H^d_{n,m}= \textbf{Im} (i_{n,m})^d_* = \frac{Z_d(K_n)}{B_d(K_m) \cap Z_d(K_n)}