Adversarially Robust Topological Inference
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866908287493996544 |
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| author | Vishwanath, Siddharth Sriperumbudur, Bharath K. Fukumizu, Kenji Kuriki, Satoshi |
| author_facet | Vishwanath, Siddharth Sriperumbudur, Bharath K. Fukumizu, Kenji Kuriki, Satoshi |
| contents | The distance function to a compact set plays a crucial role in the paradigm of topological data analysis. In particular, the sublevel sets of the distance function are used in the computation of persistent homology -- a backbone of the topological data analysis pipeline. Despite its stability to perturbations in the Hausdorff distance, persistent homology is highly sensitive to outliers. In this work, we develop a framework of statistical inference for persistent homology in the presence of outliers. Drawing inspiration from recent developments in robust statistics, we propose a \textit{median-of-means} variant of the distance function (\textsf{MoM Dist}) and establish its statistical properties. In particular, we show that, even in the presence of outliers, the sublevel filtrations and weighted filtrations induced by \textsf{MoM Dist} are both consistent estimators of the true underlying population counterpart and exhibit near minimax-optimal performance in adversarial settings. Finally, we demonstrate the advantages of the proposed methodology through simulations and applications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_01795 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Adversarially Robust Topological Inference Vishwanath, Siddharth Sriperumbudur, Bharath K. Fukumizu, Kenji Kuriki, Satoshi Statistics Theory Computational Geometry Machine Learning Algebraic Topology 62R40, 55N31, 68T09 The distance function to a compact set plays a crucial role in the paradigm of topological data analysis. In particular, the sublevel sets of the distance function are used in the computation of persistent homology -- a backbone of the topological data analysis pipeline. Despite its stability to perturbations in the Hausdorff distance, persistent homology is highly sensitive to outliers. In this work, we develop a framework of statistical inference for persistent homology in the presence of outliers. Drawing inspiration from recent developments in robust statistics, we propose a \textit{median-of-means} variant of the distance function (\textsf{MoM Dist}) and establish its statistical properties. In particular, we show that, even in the presence of outliers, the sublevel filtrations and weighted filtrations induced by \textsf{MoM Dist} are both consistent estimators of the true underlying population counterpart and exhibit near minimax-optimal performance in adversarial settings. Finally, we demonstrate the advantages of the proposed methodology through simulations and applications. |
| title | Adversarially Robust Topological Inference |
| topic | Statistics Theory Computational Geometry Machine Learning Algebraic Topology 62R40, 55N31, 68T09 |
| url | https://arxiv.org/abs/2206.01795 |