Singular Persistent Homology with Effective Concurrent Computation

Persistent homology is a popular tool in Topological Data Analysis. It provides numerical characteristics which reflect global geometric properties of data sets. In order to be useful in practice, for example for feature generation in machine learning, it needs to be effectively computable. Classical homology is a computable topological invariant because of the Mayer-Vietoris exact and spectral sequences associated to intrinsic coverings of a space. Regrettably, persistent homology itself is defined through morphing simplicial nerves of coverings of the data set. This note introduces a counterpart, the singular persistent homology, where the perspective is different. The data set is left stationary while the parameter is allowed to change in the form of the size of singular simplices. Because of this nature, coverings of the data set are much easier to handle than in other attempts to parallelize the computation of persistent homology. When computed directly, which is possible in finite metric spaces, the complexity is certainly worse than for persistent homology but not as much as one would fear. We show however that the singular and the traditional persistent homologies are isomorphic, so it is possible to perform the concurrent computations using traditional algorithms. The important point is that the advantages of distributed computation are not necessarily in speed improvement but in sheer feasibility for large data sets. 1. Definitions and the excision property Definition 1.1. The symbol ∆ will denote the metric space with n + 1 points where the distance between each pair of points is 1. Let X be a metric space. Definition 1.2. An n-dimensional singular ε-simplex σ : ∆ → X is an arbitrary set function with the diameter of the image of σ bounded by the given real number ε ≥ 0. This condition is equivalent to the requirement that σ is an ε-Lipschitz function. Let us denote the set of all such simplices by S n(X) and the vector space they generate by C n(X). The coefficients that we use in this paper are the real numbers R or a finite field, they will be implicit in the notation. A historical remark. A useful additonal condition would require that chains are locally finite in the sense that each metric ball in X intersects at most finitely many simplices in the chain. This gives the subgroup C n (X). Of course, when X is a finite metric space, there is no distinction between C n(X) and C ε,lf n (X). When X is not compact, the colimit of C n (X) is essentially the group of uniformly bounded locally finite chains that were introduced and were useful in the work on the Novikov conjecture in K-theory by G. Carlsson and the author [4].