Computing Multidimensional Persistence

Abstract

The theory of multidimensional persistence captures the topology of a multifiltration -a multiparameter family of increasing spaces. Multifiltrations arise naturally in the topological analysis of scientific data. In this paper, we give a polynomial time algorithm for computing multidimensional persistence. We recast this computation as a problem within computational commutative algebra, and utilize algorithms from this area to solve it. While the resulting problem is Expspace-complete and the standard algorithms take doubly-exponential time, we exploit the structure inherent within multifiltrations to yield practical algorithms. We implement all algorithms in the paper and provide statistical experiments to demonstrate their feasibility.

Topology and data

“…The part of that methodology which is relevant is the multigraded version of the notion of a Gröbner basis for a submodule of a free finitely generated module, the Buchberger algorithm for constructing such a basis, and the algorithm for constructing syzygies attached to homomorphisms of free multigraded modules (Schreyer's algorithm). These results are developed in [14]. The Gröbner basis provides a very compact description which contains all the information about the multidimensional persistence problem.…”

A new algorithm for computing the 2-dimensional matching distance between size functions

“…This is due to the fact that a complete, discrete and stable representation for the Persistent Topology shape descriptors seems not to be available in the multidimensional setting, differently from what happens in the 1-dimensional situation. The arising computational difficulties have been faced following different strategies [1,5,10], but not completely solved.…”

Exact Topological Inference for Paired Brain Networks via Persistent Homology

“…Then, instead of building graph filtration over 1D, we need to build it over 2D sparse parameter space. This is related to multidimensional persistent homology [4,12]. Extending the proposed method to triple image setting is left as a future study.…”