A roadmap for the computation of persistent homology

EPJ Data Science

Table 1 We summarize several types of complexes that are used for PH

Complex K	Size of K	Theoretical guarantee
Čech	\(2^{\mathcal {O}(N)}\)	Nerve theorem
Vietoris–Rips (VR)	\(2^{\mathcal {O}(N)}\)	Approximates Čech complex
Alpha	\(N^{\mathcal {O}(\lceil d/2 \rceil)}\) (N points in \(\mathbb {R}^{d}\))	Nerve theorem
Witness	\(2^{\mathcal {O}(\|L\|)}\)	For curves and surfaces in Euclidean space
Graph-induced complex	\(2^{\mathcal {O}(\|Q\|)}\)	Approximates VR complex
Sparsified Čech	\(\mathcal {O}(N)\)	Approximates Čech complex
Sparsified VR	\(\mathcal {O}(N)\)	Approximates VR complex

We indicate the theoretical guarantees and the worst-case sizes of the complexes as functions of the cardinality N of the vertex set. For the witness complexes (see Section 5.2.4), L denotes the set of landmark points, while Q denotes the subsample set for the graph-induced complex (see Section 5.2.5).