Greedy Matroid Algorithm And Computational Persistent Homology

An important problem in computational topology is to calculate the homology of a space from samples. In this work, we develop a statistical approach to this problem by calculating the expected rank of an induced map on homology from a sub-sample to the full space. We develop a greedy matroid algorithm for finding an optimal basis for the image of the induced map, and investigate the relationship between this algorithm and the probability of sampling vectors in the image of the induced map.