Topological Characterization of Global Extrema via Persistent Critical Sets
Fuente:
Zenodo
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Recurso digital |
| Veröffentlicht: |
Zenodo
2025
|
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866901803202772992 |
|---|---|
| author | SÉRGIO DE ANDRADE, PAULO |
| author_facet | SÉRGIO DE ANDRADE, PAULO |
| contents | This paper introduces a novel framework for identifying and characterizing global extrema of real-valued functions using tools from topological data analysis, specifically persistent homology. The central challenge in global optimization is distinguishing the global extremum from numerous local extrema, especially in high-dimensional or noisy settings. We propose the concept of "persistent critical sets," which are regions in the function's domain associated with long-lasting topological features in a sublevel or superlevel set filtration. By analyzing the 0-dimensional persistent homology, which tracks the birth and death of connected components, we can isolate features corresponding to significant basins of attraction or peaks. The persistence of these features serves as a robust measure of their significance, allowing for the filtering of topological noise and the identification of regions containing global extrema. We formalize the definition of persistent critical sets and provide an algorithmic approach for their computation. The method's efficacy is demonstrated on several benchmark optimization functions, such as the Rastrigin and Ackley functions, where it successfully isolates the global minimum's basin of attraction even in the presence of significant noise. This gradient-free, multiscale approach offers a new perspective on global optimization, providing not just a single point estimate but a topologically robust characterization of the regions where global extrema are located. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17606210 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Topological Characterization of Global Extrema via Persistent Critical Sets SÉRGIO DE ANDRADE, PAULO This paper introduces a novel framework for identifying and characterizing global extrema of real-valued functions using tools from topological data analysis, specifically persistent homology. The central challenge in global optimization is distinguishing the global extremum from numerous local extrema, especially in high-dimensional or noisy settings. We propose the concept of "persistent critical sets," which are regions in the function's domain associated with long-lasting topological features in a sublevel or superlevel set filtration. By analyzing the 0-dimensional persistent homology, which tracks the birth and death of connected components, we can isolate features corresponding to significant basins of attraction or peaks. The persistence of these features serves as a robust measure of their significance, allowing for the filtering of topological noise and the identification of regions containing global extrema. We formalize the definition of persistent critical sets and provide an algorithmic approach for their computation. The method's efficacy is demonstrated on several benchmark optimization functions, such as the Rastrigin and Ackley functions, where it successfully isolates the global minimum's basin of attraction even in the presence of significant noise. This gradient-free, multiscale approach offers a new perspective on global optimization, providing not just a single point estimate but a topologically robust characterization of the regions where global extrema are located. |
| title | Topological Characterization of Global Extrema via Persistent Critical Sets |
| url | https://doi.org/10.5281/zenodo.17606210 |