Persistent Betti number

Let ${\displaystyle K}$ be a simplicial complex, and let ${\displaystyle f:K\to \mathbb {R} }$ be a monotonic, i.e., non-decreasing function. Requiring monotonicity guarantees that the sublevel set ${\displaystyle K(a):=f^{-1}(-\infty ,a]}$ is a subcomplex of ${\displaystyle K}$ for all ${\displaystyle a\in \mathbb {R} }$. Letting the parameter ${\displaystyle a}$ vary, we can arrange these subcomplexes into a nested sequence ${\displaystyle \emptyset =K_{0}\subseteq K_{1}\subseteq \cdots \subseteq K_{n}=K}$ for some natural number ${\displaystyle n}$. This sequences defines a filtration on the complex ${\displaystyle K}$.

Persistent homology concerns itself with the evolution of topological features across a filtration. To that end, by taking the ${\displaystyle p^{th}}$ homology group of every complex in the filtration we obtain a sequence of homology groups ${\displaystyle 0=H_{p}(K_{0})\to H_{p}(K_{1})\to \cdots \to H_{p}(K_{n})=H_{p}(K)}$ that are connected by homomorphisms induced by the inclusion maps in the filtration. When applying homology over a field, we get a sequence of vector spaces and linear maps commonly known as a persistence module.

In order to track the evolution of homological features as opposed to the static topological information at each individual index, one needs to count only the number of nontrivial homology classes that persist in the filtration, i.e., that remain nontrivial across multiple scale parameters.

For each ${\displaystyle i\leq j}$, let ${\displaystyle f_{p}^{i,j}}$ denote the induced homomorphism ${\displaystyle H_{p}(K_{i})\to H_{p}(K_{j})}$. Then the ${\displaystyle p^{th}}$ persistent homology groups are defined to be the images of each induced map. Namely, ${\displaystyle H_{p}^{i,j}:=\operatorname {im} f_{p}^{i,j}}$ for all ${\displaystyle 0\leq i\leq j\leq n}$.

In parallel to the classical Betti number, the ${\displaystyle p^{th}}$ persistent Betti numbers are precisely the ranks of the ${\displaystyle p^{th}}$ persistent homology groups, given by the definition ${\displaystyle \beta _{p}^{i,j}:=\operatorname {rank} H_{p}^{i,j}}$.[6]