Moduli Spaces for Persistent Homology
Fuente:
Zenodo
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Recurso digital |
| Pubblicazione: |
Zenodo
2025
|
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866901472378093568 |
|---|---|
| author | Revista, Zen MATH, 10 |
| author_facet | Revista, Zen MATH, 10 |
| contents | Persistent homology, a cornerstone of Topological Data Analysis (TDA), provides a powerful framework for extracting robust topological features from complex datasets. These features are typically summarized as persistence diagrams or barcodes, which encode the birth and death times of topological phenomena across a filtered space. While persistent homology has found widespread applications, the intrinsic geometric and algebraic structure of the space of these topological summaries remains a critical area of investigation. This paper explores the concept of moduli spaces as a theoretical framework for understanding and classifying persistent homology objects. Drawing parallels from algebraic geometry, where moduli spaces parameterize and organize geometric objects with shared properties, we investigate how such a construction could provide a deeper understanding of persistent homology. We review foundational concepts of persistent homology, examine the metric geometry of persistence diagram spaces, and survey existing efforts toward defining equivalence relations and invariants that hint at underlying moduli structures. The methodology delves into how categorical approaches and metric-based definitions can contribute to constructing such spaces, discussing their potential properties and the implications for statistical analysis and machine learning with topological data. We highlight how a moduli space perspective could clarify the nature of stability results and offer novel avenues for comparing and interpreting topological signatures, thereby enriching the theoretical underpinnings and practical utility of TDA. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17807118 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Moduli Spaces for Persistent Homology Revista, Zen MATH, 10 Persistent homology, a cornerstone of Topological Data Analysis (TDA), provides a powerful framework for extracting robust topological features from complex datasets. These features are typically summarized as persistence diagrams or barcodes, which encode the birth and death times of topological phenomena across a filtered space. While persistent homology has found widespread applications, the intrinsic geometric and algebraic structure of the space of these topological summaries remains a critical area of investigation. This paper explores the concept of moduli spaces as a theoretical framework for understanding and classifying persistent homology objects. Drawing parallels from algebraic geometry, where moduli spaces parameterize and organize geometric objects with shared properties, we investigate how such a construction could provide a deeper understanding of persistent homology. We review foundational concepts of persistent homology, examine the metric geometry of persistence diagram spaces, and survey existing efforts toward defining equivalence relations and invariants that hint at underlying moduli structures. The methodology delves into how categorical approaches and metric-based definitions can contribute to constructing such spaces, discussing their potential properties and the implications for statistical analysis and machine learning with topological data. We highlight how a moduli space perspective could clarify the nature of stability results and offer novel avenues for comparing and interpreting topological signatures, thereby enriching the theoretical underpinnings and practical utility of TDA. |
| title | Moduli Spaces for Persistent Homology |
| url | https://doi.org/10.5281/zenodo.17807118 |