Persistent Homology – State of the art and challenges

A recurring task in mathematics, statistics, and computer science is understanding the connectivity information, or equivalently, the topological properties of a given object. For concreteness, we assume the object in question to be a geometric shape, possibly embedded in a high-dimensional space, although that assumption is not necessary for most of the theory. Algebraic topology offers a toolset for quantifying and comparing topological features of such shapes. The strongest notion of topological equivalence, the existence of an homeomorphism between topological spaces, is out of reach in general in computational contexts.1 An attractive compromise is offered by the theory of homology over a base field F. In informal terms, the p-th homology group Hp(S) of a shape S (with p ≥ 0) is a F-vector space whose rank counts the number of “p-dimensional holes” in S . Concretely, for objects embedded in R3, rankH0,1,2(S) count the number of connected components, tunnels, and voids, respectively, induced by the shape S . Homology over fields reveals less topological information then the Z-homology, but this partial information is sufficient for many purposes. The main advantage of restricting to fields is the existence of efficient algorithms. More precisely, if the input is given as a combinatorial cell complex, the homology groups in all dimensions can be computed in cubic time with respect to the number of cells.