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Efficient Algorithms for Complexes of Persistence Modules with Applications

Core Concepts
Efficient algorithms developed for computing homology of non-free module complexes.
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.
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.
"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."

Deeper Inquiries

How does considering relations alongside generators impact computational efficiency

Considering relations alongside generators can impact computational efficiency by allowing for a more compact representation of the modules. By incorporating relations, we can reduce redundancy and avoid unnecessary computations. This leads to a more efficient algorithm as it eliminates the need to consider all possible combinations of generators independently. Additionally, by including relations, we can capture the algebraic structure of the modules more accurately, leading to optimized computations and faster processing times.

What are potential limitations or drawbacks when dealing with complexes of non-free modules

When dealing with complexes of non-free modules, there are potential limitations and drawbacks that need to be considered. One limitation is that non-free modules may introduce additional complexity due to the presence of relations among generators. These relations may lead to increased computational overhead as handling non-trivial dependencies between generators requires more sophisticated algorithms. Furthermore, working with non-free modules could result in challenges related to basis transformations and matrix operations. The lack of a straightforward basis for these modules may complicate certain computations and make it harder to apply standard linear algebra techniques efficiently. Moreover, computing homology or cohomology in complexes of non-free modules might require specialized algorithms tailored specifically for these cases. This specialization could limit the generalizability or applicability of existing methods designed for free module complexes.

How might these algorithms be applied in other fields beyond computational geometry

These algorithms have applications beyond computational geometry in various fields such as topological data analysis (TDA), machine learning, network analysis, biology, physics, and many others where complex data structures need to be analyzed using algebraic topology concepts. In TDA applications like shape recognition or feature extraction from high-dimensional data sets, persistent homology calculations play a crucial role in understanding underlying patterns or structures present in the data. The efficient computation of persistence diagrams using these algorithms can enhance TDA methodologies significantly. In machine learning tasks such as clustering or classification problems where topological features are relevant (e.g., identifying clusters based on connectivity), persistent homology techniques enabled by these algorithms can provide valuable insights into dataset characteristics not easily captured by traditional methods. Additionally, in network analysis applications like community detection or anomaly detection within networks where understanding network connectivity patterns is essential; utilizing persistent homology tools derived from these algorithms can offer novel perspectives on network structures and behaviors.