Quantifying the structural stability of simplicial homology
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866909084044754944 |
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| author | Guglielmi, Nicola Savostianov, Anton Tudisco, Francesco |
| author_facet | Guglielmi, Nicola Savostianov, Anton Tudisco, Francesco |
| contents | The homology groups of a simplicial complex reveal fundamental properties of the topology of the data or the system and the notion of topological stability naturally poses an important yet not fully investigated question. In the current work, we study the stability in terms of the smallest perturbation sufficient to change the dimensionality of the corresponding homology group. Such definition requires an appropriate weighting and normalizing procedure for the boundary operators acting on the Hodge algebra's homology groups. Using the resulting boundary operators, we then formulate the question of structural stability as a spectral matrix nearness problem for the corresponding higher-order graph Laplacian. We develop a bilevel optimization procedure suitable for the formulated matrix nearness problem and illustrate the method's performance on a variety of synthetic quasi-triangulation datasets and transportation networks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_03627 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quantifying the structural stability of simplicial homology Guglielmi, Nicola Savostianov, Anton Tudisco, Francesco Numerical Analysis Social and Information Networks Optimization and Control Data Analysis, Statistics and Probability 05C50, 65F45, 65K10, 57M15, 62R40 The homology groups of a simplicial complex reveal fundamental properties of the topology of the data or the system and the notion of topological stability naturally poses an important yet not fully investigated question. In the current work, we study the stability in terms of the smallest perturbation sufficient to change the dimensionality of the corresponding homology group. Such definition requires an appropriate weighting and normalizing procedure for the boundary operators acting on the Hodge algebra's homology groups. Using the resulting boundary operators, we then formulate the question of structural stability as a spectral matrix nearness problem for the corresponding higher-order graph Laplacian. We develop a bilevel optimization procedure suitable for the formulated matrix nearness problem and illustrate the method's performance on a variety of synthetic quasi-triangulation datasets and transportation networks. |
| title | Quantifying the structural stability of simplicial homology |
| topic | Numerical Analysis Social and Information Networks Optimization and Control Data Analysis, Statistics and Probability 05C50, 65F45, 65K10, 57M15, 62R40 |
| url | https://arxiv.org/abs/2301.03627 |