Core Concepts

The authors prove a stability theorem for bigraded persistent double homology modules and barcodes of finite pseudo-metric spaces.

Abstract

The content discusses the following key points:
Bigraded persistent homology modules and barcodes are defined using the bigraded homology of the moment-angle complex associated with the Vietoris-Rips filtration of a finite pseudo-metric space.
Bigraded persistent homology can distinguish between point clouds that are indistinguishable by ordinary persistent homology.
The authors introduce bigraded persistent double homology, which fixes computational drawbacks of ordinary bigraded homology and has a stability property.
A stability theorem for bigraded persistent double homology is proved in two stages:
First, bigraded persistent homology is shown to satisfy a stability property with respect to a modified Gromov-Hausdorff distance.
Second, the stability of bigraded persistent double homology is established using the invariance of double homology under the doubling operation on simplicial complexes.
Examples are provided to illustrate the differences between ordinary and bigraded persistence, as well as the behavior of bigraded persistent double homology.

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Key Insights Distilled From

by Anthony Bahr... at **arxiv.org** 03-29-2024

Deeper Inquiries

The stability theorem for bigraded persistent double homology can be extended to other types of bigraded persistence modules by considering different types of filtrations or constructions. One approach could be to explore the stability properties of bigraded persistent double homology modules in the context of other types of filtrations, such as the Cech complex filtration or the witness complex filtration. By analyzing how the bigraded persistent double homology modules behave under different types of filtrations, one can generalize the stability theorem to a broader range of scenarios. Additionally, exploring the stability of bigraded persistence modules in the context of different topological spaces or data structures can also provide insights into the generalizability of the stability theorem.

The potential applications of bigraded persistent double homology in data analysis and topology are vast and diverse. In data analysis, bigraded persistent double homology can be used to capture more nuanced and detailed topological features of complex data sets. This can lead to improved clustering algorithms, anomaly detection methods, and shape recognition techniques. By leveraging the bigraded structure of persistent homology, researchers and practitioners can gain deeper insights into the geometric and topological properties of data sets, enabling more accurate and robust analysis.
In topology, bigraded persistent double homology can be applied to study the homological properties of spaces with intricate structures and symmetries. It can help in detecting subtle topological changes, distinguishing between different types of spaces, and understanding the underlying geometric relationships between data points. Furthermore, the bigraded structure of persistent homology can provide a richer representation of topological spaces, allowing for more sophisticated topological invariants and classifications.

The bigraded structure of persistent homology reflects a deeper connection between the algebraic properties of the homology modules and the geometric or topological characteristics of the data set. The bidegrees in bigraded persistent homology encode information about the dimensions and multiplicities of topological features present in the data, offering a more detailed and nuanced representation of the underlying space.
By analyzing the bigraded structure of persistent homology, researchers can uncover hidden patterns, symmetries, and complexities in the data set's topology. The bidegrees provide a systematic way to organize and interpret the homological information, shedding light on the spatial relationships, connectivity, and shape properties of the data points. This deeper connection between the bigraded structure of persistent homology and the underlying geometry or topology enhances the analytical power and interpretability of topological data analysis methods.

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