Core Concepts
Efficient algorithms developed for computing homology of non-free module complexes.
Abstract
The article introduces efficient algorithms for computing the homology of complexes of persistence modules that are not free. It extends the traditional persistence algorithm to handle presentations of modules and compute homology efficiently. The methods presented enable the computation of persistent (co)homology in various scenarios, such as simplicial towers and cosheaves over simplicial complexes. The paper also addresses the computation of cohomology over finite posets by reducing it to computations over simplicial complexes. The approach involves handling relations in addition to generators, leading to the design of algorithms that can manage non-free modules efficiently.
Stats
We replace persistence modules by their presentations and develop an efficient algorithm to compute the homology of a complex of presentations.
Our method considers a generator only once even though it may exist over a wide range of indices.
We can compute the homology of a complex of non-free modules if we consider relations in addition to the generators.
The closest work along this line is recent research where barcode bases and operations on them are introduced to compute a barcode basis efficiently.
Preliminary implementation results suggest practical utility beyond theoretical implications.
Quotes
"We extend the persistence algorithm, viewed as an algorithm computing the homology of a complex."
"Our methods lead to a new efficient algorithm for computing persistent homology."
"We tackle the problem directly at the algebraic level by designing algorithms that can handle complexes of non-free modules."
"One of our main observations is that we can compute the homology of a complex of non-free modules if we consider relations in addition to the generators."
"We have already mentioned that our approach provides a novel efficient algorithm."