In topological data analysis, persistent homology is a method for computing topological features of a space at different spatial resolutions. More persistent features are detected over a wide range of spatial scales and are deemed more likely to represent true features of the underlying space rather than artifacts of sampling, noise, or particular choice of parameters. To find the persistent homology of a space, the space must first be represented as a simplicial complex. A distance function on the underlying space corresponds to a filtration of the simplicial complex, that is a nested sequence of increasing subsets. One common method of doing this is via taking the sublevel filtration of the distance to a point cloud, or equivalently, the offset filtration on the point cloud and taking its nerve in order to get the simplicial filtration known as Čech filtration. A similar construction uses a nested sequence of Vietoris–Rips complexes known as the Vietoris–Rips filtration.
Definition Formally, consider a real-valued function on a simplicial complex f : K → R {\displaystyle f:K\rightarrow \mathbb {R} } that is non-decreasing on increasing sequences of faces, so f ( σ ) ≤ f ( τ ) {\displaystyle f(\sigma )\leq f(\tau )} whenever σ {\displaystyle \sigma } is a face of τ {\displaystyle \tau } in K {\displaystyle K} . Then for every a ∈ R {\displaystyle a\in \mathbb {R} } the sublevel set K a = f − 1 ( ( − ∞ , a ] ) {\displaystyle K_{a}=f^{-1}((-\infty ,a])} is a subcomplex of K, and the ordering of the values of f {\displaystyle f} on the simplices in K {\displaystyle K} (which is in practice always finite) induces an ordering on the sublevel complexes that defines a filtration
∅ = K 0 ⊆ K 1 ⊆ ⋯ ⊆ K n = K {\displaystyle \emptyset =K_{0}\subseteq K_{1}\subseteq \cdots \subseteq K_{n}=K}
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