Gregory Henselman-Petrusek

Jiajie Luo, Gregory Henselman-Petrusek

We study pointwise free and finitely-generated persistence modules over a principal ideal domain, indexed by a (possibly infinite) totally-ordered poset category. We show that such persistence modules admit interval decompositions if and only if every structure map has free cokernel. We also show that, in torsion-free settings, the integer persistent homology module of a filtration of topological spaces admits an interval decomposition if and only if the associated persistence diagram is invariant to the choice of coefficient field. These results generalize prior work where the indexing category is finite.

Lu Li, Connor Thompson, Gregory Henselman-Petrusek et al.

Cycle representatives of persistent homology classes can be used to provide descriptions of topological features in data. However, the non-uniqueness of these representatives creates ambiguity and can lead to many different interpretations of the same set of classes. One approach to solving this problem is to optimize the choice of representative against some measure that is meaningful in the context of the data. In this work, we provide a study of the effectiveness and computational cost of several $\ell_1$-minimization optimization procedures for constructing homological cycle bases for persistent homology with rational coefficients in dimension one, including uniform-weighted and length-weighted edge-loss algorithms as well as uniform-weighted and area-weighted triangle-loss algorithms. We conduct these optimizations via standard linear programming methods, applying general-purpose solvers to optimize over column bases of simplicial boundary matrices. Our key findings are: (i) optimization is effective in reducing the size of cycle representatives, (ii) the computational cost of optimizing a basis of cycle representatives exceeds the cost of computing such a basis in most data sets we consider, (iii) the choice of linear solvers matters a lot to the computation time of optimizing cycles, (iv) the computation time of solving an integer program is not significantly longer than the computation time of solving a linear program for most of the cycle representatives, using the Gurobi linear solver, (v) strikingly, whether requiring integer solutions or not, we almost always obtain a solution with the same cost and almost all solutions found have entries in {-1, 0, 1} and therefore, are also solutions to a restricted $\ell_0$ optimization problem, and (vi) we obtain qualitatively different results for generators in Erdős-Rényi random clique complexes.