Numeric Invariants from Multidimensional Persistence

We extend the results of Adcock, Carlsson, and Carlsson ([ACC13]) by constructing numeric invariants from the computation of a multidimensional persistence module as given by Carlsson, Singh, and Zomorodian in [CSZ10]. The use of topology to study point cloud data has been well established ([Car09], [Car14]). Given a finite metric space (e.g., a finite set in R n), one first constructs a filtered complex which attempts to approximate the shape of the underlying data. Commonly used complexes include the Vietoris-Rips complex, the Cech complex, the α-complex, and the witness complex. The persistent homology of a filtered complex is an abstract algebraic entity which combines information about the ho-mology of the levelwise complexes in a filtered simplicial complex and the maps on homology induced by the filtration maps of the complex. The process of constructing a filtered complex and taking persistent homology provides abstract algebraic information about the original point cloud data. It is difficult to interpret raw calculations of persistent homology from a geometric and intuitive standpoint. This is partially remedied by Adcock, Carlsson, and Carlsson, who have successfully studied ways of interpreting persistent homology geometrically through the construction of numeric invariants [ACC13]. They produce an infinite family of functions, each of which takes as input any one-dimensional persistence module (the most notable of such objects being the persistent homology of a one-dimensional filtered complex) and outputs a nonnegative number which has a concrete interpretation in terms of the geometric features of the filtered complex. If the filtered complex was obtained from point cloud data, these values provide information about the size and density of the geometric features in the point cloud data. The process by which one obtains values from point cloud data by constructing a filtered complex, taking homology, and calculating functions serves as a useful pipeline for the construction of numerical features which can then be used in machine learning algorithms. Unfortunately, the method of [ACC13] does not generalize nicely to multidimen-sional persistence modules. The construction of the algebraic functions in [ACC13] relies heavily on the classification theorem of finitely generated modules over a PID and the categorical equivalence between one-dimensional persistence modules and finitely generated modules over a PID. Multidimensional persistence modules are categorically equivalent to finitely presented graded R[x 1 , ..., x n ]-modules, for which there is no analogous classification theorem. Furthermore, the results of [CZ09] show that there is no complete discrete …