2000 character limit reached
Persistent sheaf Laplacians
Published 20 Dec 2021 in math.AT | (2112.10906v4)
Abstract: Recently various types of topological Laplacians have been studied from the perspective of data analysis. The spectral theory of these Laplacians has significantly extended the scope of algebraic topology and data analysis. Inspired by the theory of persistent Laplacians and cellular sheaves, this work develops the theory of persistent sheaf Laplacians for cellular sheaves, and describes how to construct sheaves for a point cloud where each point is associated with a quantity that can be devised to embed physical properties. As a result, the spectra of persistent sheaf Laplacians encode both geometrical and non-geometrical information of the given point cloud. The theory of persistent sheaf Laplacians is an elegant method for fusing different types of data and has huge potential for future development.
- Quantum persistent homology. arXiv preprint arXiv:2202.12965, 2022.
- N. Berkouk and G. Ginot. A derived isometry theorem for constructible sheaves on ℝℝ\mathbb{R}blackboard_R, 2021.
- P. Bubenik and N. Milićević. Homological algebra for persistence modules. Foundations of Computational Mathematics, Jan 2021.
- Z. Cang and G.-W. Wei. Topologynet: Topology based deep convolutional and multi-task neural networks for biomolecular property predictions. PLoS computational biology, 13(7):e1005690, 2017.
- Z. Cang and G.-W. Wei. Persistent cohomology for data with multicomponent heterogeneous information. SIAM Journal on Mathematics of Data Science, 2(2):396–418, 2020.
- Betti numbers in multidimensional persistent homology are stable functions. Mathematical Methods in the Applied Sciences, 36(12):1543–1557, 2013.
- Persistent laplacian projected omicron ba. 4 and ba. 5 to become new dominating variants. Computers in Biology and Medicine, 151:106262, 2022.
- Evolutionary de rham-hodge method. arXiv preprint arXiv:1912.12388, 2019.
- F. R. Chung. Spectral graph theory, volume 92. American Mathematical Soc., 1997.
- J. Curry. Sheaves, cosheaves and applications. PhD thesis, University of Pennsylvania, 2014.
- J. Dodziuk. de rham-hodge theory for l2-cohomology of infinite coverings. Topology, 16(2):157–165, 1977.
- Persistent homology-a survey. Contemporary mathematics, 453:257–282, 2008.
- R. Ghrist. Elementary Applied Topology. Createspace, 1 edition, 2014.
- Bacteriocin as-48, a microbial cyclic polypeptide structurally and functionally related to mammalian nk-lysin. Proceedings of the National Academy of Sciences, 97(21):11221–11226, 2000.
- The laplacian spectrum of a graph. SIAM Journal on matrix analysis and applications, 11(2):218–238, 1990.
- H. Hang and W. Mio. Correspondence modules and persistence sheaves: A unifying perspective on one-parameter persistent homology, 2021.
- J. Hansen and R. Ghrist. Toward a spectral theory of cellular sheaves. Journal of Applied and Computational Topology, 3(4):315–358, 2019.
- Improvements to the apbs biomolecular solvation software suite. Protein science : a publication of the Protein Society, 27(1):112–128, Jan 2018. 28836357[pmid].
- Computational homology, volume 3. Springer, 2004.
- L.-H. Lim. Hodge laplacians on graphs. Siam Review, 62(3):685–715, 2020.
- The algebraic stability for persistent laplacians. arXiv preprint arXiv:2302.03902, 2023.
- Persistent laplacians: Properties, algorithms and implications. SIAM Journal on Mathematics of Data Science, 4(2):858–884, 2022.
- Weighted persistent homology for biomolecular data analysis. Scientific reports, 10(1):2079, 2020.
- Z. Meng and K. Xia. Persistent spectral–based machine learning (perspect ml) for protein-ligand binding affinity prediction. Science Advances, 7(19):eabc5329, 2021.
- J. R. Munkres. Elements of Algebraic Topology. Addison-Wesley Publishing Company, Inc., 1984.
- A review of mathematical representations of biomolecular data. Physical Chemistry Chemical Physics, 22(8):4343–4367, 2020.
- Mathematical deep learning for pose and binding affinity prediction and ranking in d3r grand challenges. Journal of computer-aided molecular design, 33(1):71–82, 2019.
- Mathdl: mathematical deep learning for d3r grand challenge 4. Journal of computer-aided molecular design, 34(2):131–147, 2020.
- Y. Qiu and G.-W. Wei. Persistent spectral theory-guided protein engineering. Nature Computational Science, 3(2):149–163, 2023.
- Atomic charge calculator ii: web-based tool for the calculation of partial atomic charges. Nucleic acids research, 48(W1):W591–W596, 2020.
- M. Robinson. How do we deal with noisy data?
- M. Robinson. Topological Signal Processing. Springer, Berlin, Heidelberg, 1 edition, 2014.
- F. Russold. Persistent sheaf cohomology. arXiv preprint arXiv:2204.13446, 2022.
- Samplchallenges. Sampl6/cb8.mol2 at master · samplchallenges/sampl6.
- A. D. Shepard. A cellular description of the derived category of a stratified space. PhD thesis, Brown University, 1985.
- Representation of molecular structures with persistent homology for machine learning applications in chemistry. Nature communications, 11(1):1–9, 2020.
- Persistent spectral graph. International Journal for Numerical Methods in Biomedical Engineering, 36(9):e3376, 2020.
- Hermes: Persistent spectral graph software. Foundations of Data Science, 3(1):67–97, 2020.
- L. Wasserman. Topological data analysis. Annual Review of Statistics and Its Application, 5:501–532, 2018.
- Weighted (co)homology and weighted laplacian. arXiv preprint arXiv:1804.06990, 2018.
- K. Yegnesh. Persistence and sheaves. arXiv preprint arXiv:1612.03522, 2016.
- The de rham–hodge analysis and modeling of biomolecules. Bulletin of mathematical biology, 82(8):1–38, 2020.
- A. Zomorodian and G. Carlsson. Computing persistent homology. Discrete & Computational Geometry, 33(2):249–274, 2005.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.