| _version_ | 1866901190974898176 |
|---|---|
| author | SÉRGIO DE ANDRADE, PAULO |
| author_facet | SÉRGIO DE ANDRADE, PAULO |
| contents | One-parameter persistent homology is a cornerstone of topological data analysis, offering a stable and complete invariant: the persistence barcode. The extension to multi-parameter persistent homology, while natural for complex data, is hindered by the absence of a simple barcode decomposition, a consequence of the underlying algebra's wild representation type. This has obstructed the development of robust, interpretable invariants. This paper confronts this challenge by introducing a framework to compute stable, barcode-like summaries for multi-parameter persistence modules. Our approach avoids direct module decomposition. Instead, we systematically project the multi-parameter module onto a canonical set of lines, generating a collection of one-parameter modules, each with a well-defined, stable barcode. The novelty of our work lies in a method to aggregate these barcodes into a single, comprehensive feature vector and, crucially, in providing a formal stability guarantee for this summary. We establish a stability theorem proving that our invariant is continuous with respect to the interleaving distance on multi-parameter modules. This result ensures robustness to noise and provides a theoretically sound, computationally tractable method for applying multi-parameter topological analysis to real-world data. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17681596 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Stable Barcodes for Multi-Parameter Persistent Homology SÉRGIO DE ANDRADE, PAULO One-parameter persistent homology is a cornerstone of topological data analysis, offering a stable and complete invariant: the persistence barcode. The extension to multi-parameter persistent homology, while natural for complex data, is hindered by the absence of a simple barcode decomposition, a consequence of the underlying algebra's wild representation type. This has obstructed the development of robust, interpretable invariants. This paper confronts this challenge by introducing a framework to compute stable, barcode-like summaries for multi-parameter persistence modules. Our approach avoids direct module decomposition. Instead, we systematically project the multi-parameter module onto a canonical set of lines, generating a collection of one-parameter modules, each with a well-defined, stable barcode. The novelty of our work lies in a method to aggregate these barcodes into a single, comprehensive feature vector and, crucially, in providing a formal stability guarantee for this summary. We establish a stability theorem proving that our invariant is continuous with respect to the interleaving distance on multi-parameter modules. This result ensures robustness to noise and provides a theoretically sound, computationally tractable method for applying multi-parameter topological analysis to real-world data. |
| title | Stable Barcodes for Multi-Parameter Persistent Homology |
| url | https://doi.org/10.5281/zenodo.17681596 |