Grassmannian Persistence Diagrams: Special Properties in the 1-Parameter Setting
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| Format: | Preprint |
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2025
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| _version_ | 1866908342205546496 |
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| author | Gülen, Aziz Burak Mémoli, Facundo Wan, Zhengchao |
| author_facet | Gülen, Aziz Burak Mémoli, Facundo Wan, Zhengchao |
| contents | In this paper, we explore the discriminative power of Grassmannian persistence diagrams of 1-parameter filtrations, examine their relationships with other related constructions, and study their computational aspects. Grassmannian persistence diagrams are defined through Orthogonal Inversion, a notion analogous to Möbius inversion. We focus on the behavior of this inversion for the poset of segments of a linear poset. We demonstrate how Grassmannian persistence diagrams of 1-parameter filtrations are connected to persistent Laplacians via a variant of orthogonal inversion tailored for the reverse-inclusion order on the poset of segments. Additionally, we establish an explicit isomorphism between Grassmannian persistence diagrams and Harmonic Barcodes via a projection. Finally, we show that degree-0 Grassmannian persistence diagrams are equivalent to treegrams, a generalization of dendrograms. Consequently, we conclude that finite ultrametric spaces can be recovered from the degree-0 Grassmannian persistence diagram of their Vietoris-Rips filtrations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_06077 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Grassmannian Persistence Diagrams: Special Properties in the 1-Parameter Setting Gülen, Aziz Burak Mémoli, Facundo Wan, Zhengchao Combinatorics Algebraic Topology In this paper, we explore the discriminative power of Grassmannian persistence diagrams of 1-parameter filtrations, examine their relationships with other related constructions, and study their computational aspects. Grassmannian persistence diagrams are defined through Orthogonal Inversion, a notion analogous to Möbius inversion. We focus on the behavior of this inversion for the poset of segments of a linear poset. We demonstrate how Grassmannian persistence diagrams of 1-parameter filtrations are connected to persistent Laplacians via a variant of orthogonal inversion tailored for the reverse-inclusion order on the poset of segments. Additionally, we establish an explicit isomorphism between Grassmannian persistence diagrams and Harmonic Barcodes via a projection. Finally, we show that degree-0 Grassmannian persistence diagrams are equivalent to treegrams, a generalization of dendrograms. Consequently, we conclude that finite ultrametric spaces can be recovered from the degree-0 Grassmannian persistence diagram of their Vietoris-Rips filtrations. |
| title | Grassmannian Persistence Diagrams: Special Properties in the 1-Parameter Setting |
| topic | Combinatorics Algebraic Topology |
| url | https://arxiv.org/abs/2504.06077 |