Generating High Dimensional Test Data for Topological Data Analysis

Topological Data Analysis (TDA) characterizes data based on topological invariants present in the data. In general, TDA treats the data as a discrete sampling of an underlying manifold. While based in the field of topology, TDA is primarily vested in the three computational elements: Persistent Homology, Euler Characteristic, and mapper. The focus of this paper is on developing infrastructure to generate synthetic test data suitable to evaluate computational elements of TDA. The objective of this work is to generate test data with known topological invariants. While it is possible to use test data of known topological objects such as n-spheres and n-tori, these structures present limited opportunities to fully exercise TDA tools. This is especially true for high-dimensional and big data. This work supports the generation of test data with tools that use algebraic expressions of manifold structures to sample the data. The approach is augmented with additional tools to combine test data sets (possibly from various dimensions \(n_i\) ) into an ambient dimension k ( \(k \ge n_i\) ) with rotations. The motivation for this work is to support verification of algorithms to implement TDA computational elements.