Betti Curves of Rank One Symmetric Matrices

Carina Curto; Joshua Paik; Igor Rivin

doi:10.1007/978-3-030-80209-7_69

Betti Curves of Rank One Symmetric Matrices

Carina Curto, Joshua Paik, Igor Rivin

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

4 Scopus citations

Abstract

Betti curves of symmetric matrices were introduced in [3] as a new class of matrix invariants that depend only on the relative ordering of matrix entries. These invariants are computed using persistent homology, and can be used to detect underlying structure in biological data that may otherwise be obscured by monotone nonlinearities. Here we prove three theorems that characterize the Betti curves of rank 1 symmetric matrices. We then illustrate how these Betti curve signatures arise in natural data obtained from calcium imaging of neural activity in zebrafish.

Original language	English (US)
Title of host publication	Geometric Science of Information - 5th International Conference, GSI 2021, Proceedings
Editors	Frank Nielsen, Frédéric Barbaresco
Publisher	Springer Science and Business Media Deutschland GmbH
Pages	645-655
Number of pages	11
ISBN (Print)	9783030802080
DOIs	https://doi.org/10.1007/978-3-030-80209-7_69
State	Published - 2021
Event	5th International Conference on Geometric Science of Information, GSI 2021 - Paris, France Duration: Jul 21 2021 → Jul 23 2021

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	12829 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	5th International Conference on Geometric Science of Information, GSI 2021
Country/Territory	France
City	Paris
Period	7/21/21 → 7/23/21

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
General Computer Science

Access to Document

10.1007/978-3-030-80209-7_69

Cite this

Curto, C., Paik, J., & Rivin, I. (2021). Betti Curves of Rank One Symmetric Matrices. In F. Nielsen, & F. Barbaresco (Eds.), Geometric Science of Information - 5th International Conference, GSI 2021, Proceedings (pp. 645-655). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 12829 LNCS). Springer Science and Business Media Deutschland GmbH. https://doi.org/10.1007/978-3-030-80209-7_69

Curto, Carina ; Paik, Joshua ; Rivin, Igor. / Betti Curves of Rank One Symmetric Matrices. Geometric Science of Information - 5th International Conference, GSI 2021, Proceedings. editor / Frank Nielsen ; Frédéric Barbaresco. Springer Science and Business Media Deutschland GmbH, 2021. pp. 645-655 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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abstract = "Betti curves of symmetric matrices were introduced in [3] as a new class of matrix invariants that depend only on the relative ordering of matrix entries. These invariants are computed using persistent homology, and can be used to detect underlying structure in biological data that may otherwise be obscured by monotone nonlinearities. Here we prove three theorems that characterize the Betti curves of rank 1 symmetric matrices. We then illustrate how these Betti curve signatures arise in natural data obtained from calcium imaging of neural activity in zebrafish.",

author = "Carina Curto and Joshua Paik and Igor Rivin",

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Curto, C, Paik, J & Rivin, I 2021, Betti Curves of Rank One Symmetric Matrices. in F Nielsen & F Barbaresco (eds), Geometric Science of Information - 5th International Conference, GSI 2021, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 12829 LNCS, Springer Science and Business Media Deutschland GmbH, pp. 645-655, 5th International Conference on Geometric Science of Information, GSI 2021, Paris, France, 7/21/21. https://doi.org/10.1007/978-3-030-80209-7_69

Betti Curves of Rank One Symmetric Matrices. / Curto, Carina; Paik, Joshua; Rivin, Igor.
Geometric Science of Information - 5th International Conference, GSI 2021, Proceedings. ed. / Frank Nielsen; Frédéric Barbaresco. Springer Science and Business Media Deutschland GmbH, 2021. p. 645-655 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 12829 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

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N2 - Betti curves of symmetric matrices were introduced in [3] as a new class of matrix invariants that depend only on the relative ordering of matrix entries. These invariants are computed using persistent homology, and can be used to detect underlying structure in biological data that may otherwise be obscured by monotone nonlinearities. Here we prove three theorems that characterize the Betti curves of rank 1 symmetric matrices. We then illustrate how these Betti curve signatures arise in natural data obtained from calcium imaging of neural activity in zebrafish.

AB - Betti curves of symmetric matrices were introduced in [3] as a new class of matrix invariants that depend only on the relative ordering of matrix entries. These invariants are computed using persistent homology, and can be used to detect underlying structure in biological data that may otherwise be obscured by monotone nonlinearities. Here we prove three theorems that characterize the Betti curves of rank 1 symmetric matrices. We then illustrate how these Betti curve signatures arise in natural data obtained from calcium imaging of neural activity in zebrafish.

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Curto C, Paik J, Rivin I. Betti Curves of Rank One Symmetric Matrices. In Nielsen F, Barbaresco F, editors, Geometric Science of Information - 5th International Conference, GSI 2021, Proceedings. Springer Science and Business Media Deutschland GmbH. 2021. p. 645-655. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-030-80209-7_69

Betti Curves of Rank One Symmetric Matrices

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