Persistent Stanley–Reisner theory

Faisal Suwayyid; Guo-Wei Wei

doi:10.3934/fods.2025009

Article Contents

2026, Volume 8: 287-312. Doi: 10.3934/fods.2025009

This volume Previous Article Curse of dimensionality on persistence diagrams Next Article

Persistent Stanley–Reisner theory

Faisal Suwayyid^{1, 2,} and
Guo-Wei Wei^{2, 3, 4, ,}

1.
Department of Mathematics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Kingdom of Saudi Arabia
2.
Department of Mathematics, Michigan State University, MI 48824, USA
3.
Department of Electrical and Computer Engineering, Michigan State University, MI 48824, USA
4.
Department of Biochemistry and Molecular Biology, Michigan State University, MI 48824, USA

^*Corresponding author: Guo-Wei Wei
^*Corresponding author: Guo-Wei Wei

Received: March 2025

Revised: May 2025

Early access: June 10, 2025

Published online: June 10, 2025

Abstract / Introduction Full Text(HTML) Figure(8) / Table(2) Related Papers Cited by

Abstract

Topological data analysis (TDA) has emerged as an effective approach in data science, with its key technique, persistent homology, rooted in algebraic topology. Although alternative approaches based on differential topology, geometric topology, and combinatorial Laplacians have been proposed, combinatorial commutative algebra has hardly been developed for machine learning and data science. In this work, we introduce persistent Stanley–Reisner theory to bridge commutative algebra, combinatorial algebraic topology, machine learning, and data science. We propose persistent h-vectors, persistent f-vectors, persistent graded Betti numbers, persistent facet ideals, and facet persistence modules. Stability analysis indicates that these algebraic invariants are stable against geometric perturbations. We carried out Stanley–Reisner machine learning prediction of a molecular dataset to demonstrate the utility of the proposed persistent Stanley–Reisner theory for practical applications.

Keywords:

Mathematics Subject Classification: Primary: 62R40; Secondary: 55N31.

Citation:

Full Text(HTML)

Figure 1. A geometric representation of six vertices forming a pyramid with two distinct loops, each attached to different edges

Download: Full-size image PowerPoint slide

Figure 2. Three representative plots of the filtration of a triangular bipyramid with an equatorial triangular cross-section

Download: Full-size image PowerPoint slide

Figure 3. An illustration of the persistent variations of the $ h $-vectors, $ f $-vectors, Betti numbers, and graded Betti numbers, respectively. Each curve represents a function that assumes discrete integer values. To enhance the visual clarity of the plots and prevent overlap, a slight increment is added to the curves, ensuring that each curve remains distinguishable within the graphical representation

Download: Full-size image PowerPoint slide

Figure 4. Illustration of the methodology and key steps in the proposed application of persistent Stanley–Reisner theory. Given a molecular input, a corresponding simplicial complex with an associated filtration is generated. Critical values and facet persistence barcodes are computed, leading to the construction of relevant features

Download: Full-size image PowerPoint slide

Figure 5. Structural representations and filtrations of two isomers. The top row presents the first isomer and its corresponding filtration, while the bottom row depicts the second and filtration processes. In both structures, hydrogen atoms are represented in white, boron atoms in green, and carbon atoms in brown

Download: Full-size image PowerPoint slide

Figure 6. Facet persistence barcodes of the two isomers. (a) the barcodes of the first isomer. (b) the barcodes of the second isomer. The first isomer exhibits a single essential persistent edge, whereas the second isomer features at least three or more persistent edges

Download: Full-size image PowerPoint slide

Figure 7. Facet persistence diagrams for the isomers under investigation. (a) the diagram corresponding to the first isomer, capturing the persistence of topological features across the first three homological dimensions. (b) the diagram corresponding to the second isomer, similarly capturing the persistence intervals of the faces of the first three dimensions

Download: Full-size image PowerPoint slide

Figure 8. The performance of utilizing the critical values of the persistent facet ideals evaluated with six metrics in classifying OIHP materials. The 5-Nearest Neighbors (5-NN) classifier is used on the sets of the critical values generated from the Vietoris–Rips filtration

Download: Full-size image PowerPoint slide

Table 1. Table of graded Betti numbers of the simplicial complex in Figure 1

	0	1	2	3	4
0	1	0	0	0	0
1	0	5	6	2	0
2	0	2	6	6	2
3	0	1	2	1	0

| Show Table

DownLoad: CSV

Table 2. From left to right: The betti tables at the critical values of the filtration in Figure 2

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	H. Adams, T. Emerson, M. Kirby, R. Neville, C. Peterson, P. Shipman, S. Chepushtanova, E. Hanson, F. Motta and L. Ziegelmeier, Persistence images: A stable vector representation of persistent homology, Journal of Machine Learning Research, 18 (2017), 1-35.
[2]	B. Ameneyro, V. Maroulas and G. Siopsis, Quantum persistent homology, Journal of Applied and Computational Topology, 8 (2024), 1961-1980. doi: 10.1007/s41468-023-00160-7.
[3]	D. V. Anand, Q. Xu, J. Wee, K. Xia and T. C. Sum, Topological feature engineering for machine learning based halide perovskite materials design, npj Computational Materials, 8 (2022), 203. doi: 10.1038/s41524-022-00883-8.
[4]	M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Series in Mathematics. Addison-Wesley Publishing Company, Reading, MA, 1969.
[5]	W. Bruns and J. Herzog, Cohen–Macaulay Rings, volume 39 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9780511608681.
[6]	P. Bubenik, Statistical topological data analysis using persistence landscapes, Journal of Machine Learning Research, 16 (2015), 77-102.
[7]	Z. Cang, L. Mu, K. Wu, K. Opron, K. Xia and G.-W. Wei, A topological approach for protein classification, Computational and Mathematical Biophysics, 3 (2015), 140-162. doi: 10.1515/mlbmb-2015-0009.
[8]	Z. Cang and G.-W. Wei, TopologyNet: Topology based deep convolutional and multi-task neural networks for biomolecular property predictions, PLoS Computational Biology, 13 (2017), e1005690. doi: 10.1371/journal.pcbi.1005690.
[9]	F. Chazal, V. de Silva, M. Glisse and S. Oudot, The Structure and Stability of Persistence Modules, Springer Briefs in Mathematics, 2016. doi: 10.1007/978-3-319-42545-0.
[10]	D. Chen, J. Liu, J. Wu, G.-W. Wei, F. Pan and S.-T. Yau, Path topology in molecular and materials sciences, The Journal of Physical Chemistry Letters, 14 (2023), 954-964. doi: 10.1021/acs.jpclett.2c03706.
[11]	J. Chen, R. Wang, M. Wang and G.-W. Wei, Mutations strengthened SARS-CoV-2 infectivity, Journal of Molecular Biology, 432 (2020), 5212-5226. doi: 10.1016/j.jmb.2020.07.009.
[12]	J. Chen, R. Zhao, Y. Tong and G.-W. Wei, Evolutionary de rham–hodge method, Discrete and Continuous Dynamical Systems - Series B, 26 (2021), 3785-3821. doi: 10.3934/dcdsb.2020257.
[13]	J. Clayden, N. Greeves and S. Warren, Organic Chemistry, 2 edition, Oxford University Press, Oxford, 2012. doi: 10.1093/hesc/9780199270293.001.0001.
[14]	H. Edelsbrunner and J. Harer, Persistent homology—a survey, In Surveys on Discrete and Computational Geometry: Twenty Years Later, volume 453 of Contemporary Mathematics. American Mathematical Society, 2008,257-282. doi: 10.1090/conm/453/08802.
[15]	D. Eisenbud, Commutative Algebra: With a View Toward Algebraic Geometry, volume 150 of Graduate Texts in Mathematics, Springer Science & Business Media, New York, 2013. doi: 10.1007/978-1-4612-5350-1.
[16]	R. Ghrist, Barcodes: The persistent topology of data, Bulletin of the American Mathematical Society, 45 (2008), 61-75. doi: 10.1090/S0273-0979-07-01191-3.
[17]	H. T. Hà and A. Van Tuyl, Monomial ideals, edge ideals of hypergraphs, and their graded betti numbers, Journal of Algebraic Combinatorics, 27 (2008), 215-245. doi: 10.1007/s10801-007-0079-y.
[18]	F. Hensel, M. Moor and B. Rieck, A survey of topological machine learning methods, Frontiers in Artificial Intelligence, 4 (2021). doi: 10.3389/frai.2021.681108.
[19]	G. Hetyei, On the stanley ring of cubical complex, Discrete & Computational Geometry, 14 (1995), 305-330. doi: 10.1007/BF02570709.
[20]	F. A. Khasawneh, E. Munch, D. Barnes, et al., Teaspoon: A python package for topological signal processing, Journal of Open Source Software, 10 (2025), 7243. doi: 10.21105/joss.07243.
[21]	J. Liu, J. Li and J. Wu, The algebraic stability for persistent laplacians, Homology, Homotopy and Applications, 26 (2024), 297-323. doi: 10.4310/HHA.2024.v26.n2.a15.
[22]	X. Liu, H. Feng, J. Wu and K. Xia, Persistent spectral hypergraph based machine learning (psh-ml) for protein–ligand binding affinity prediction, Briefings in Bioinformatics, 22 (2021), bbab127. doi: 10.1093/bib/bbab127.
[23]	F. Mémoli, Z. Wan and Y. Wang, Persistent laplacians: Properties, algorithms and implications, SIAM Journal on Mathematics of Data Science, 4 (2022), 858-884. doi: 10.1137/21M1435471.
[24]	E. Miller and B. Sturmfels, Combinatorial Commutative Algebra, volume 227 of Graduate Texts in Mathematics, Springer, 2005.
[25]	E. C. Mitchell, B. Story, D. Boothe, P. J. Franaszczuk and V. Maroulas, A topological deep learning framework for neural spike decoding, Biophysical Journal, 123 (2024), 2781-2789. doi: 10.1016/j.bpj.2024.01.025.
[26]	D. L. Nelson and M. M. Cox, Lehninger Principles of Biochemistry, W. H. Freeman, New York, 7 edition, 2017.
[27]	D. D. Nguyen, Z. Cang, K. Wu, M. Wang, Y. Cao and G.-W. Wei, Mathematical deep learning for pose and binding affinity prediction and ranking in D3R grand challenges, Journal of Computer-Aided Molecular Design, 33 (2019), 71-82. doi: 10.1007/s10822-018-0146-6.
[28]	T. Papamarkou, T. Birdal, M. Bronstein, G. Carlsson, J. Curry, Y. Gao, M. Hajij, R. Kwitt, P. Lio, P. Di Lorenzo, et al., Position: Topological deep learning is the new frontier for relational learning, Proceedings of Machine Learning Research, 235 (2024), 39529-39555.
[29]	L. Pauling, The nature of the chemical bond, Journal of the American Chemical Society, 67 (1945), 546-551.
[30]	J. A. Perea, Topological time series analysis, Notices of the American Mathematical Society, 66 (2019), 686-694. doi: 10.1090/noti1869.
[31]	L. Shen, J. Liu and G.-W. Wei, Evolutionary Khovanov homology, AIMS Mathematics, 9 (2024), 26139-26165. doi: 10.3934/math.20241277.
[32]	L. Shen, J. Liu and G.-W. Wei, Persistent mayer homology and persistent mayer laplacian, Foundations of Data Science, 6 (2024), 584-612. doi: 10.3934/fods.2024032.
[33]	R. P. Stanley, Combinatorics and Commutative Algebra, volume 41 of Progress in Mathematics, Birkhäuser, Boston, MA, 2 edition, 1996.
[34]	F. Suwayyid and G.-W. Wei, Persistent mayer dirac, Journal of Physics: Complexity, 5 (2024), 045005. doi: 10.1088/2632-072X/ad83a5.
[35]	F. Suwayyid and G.-W. Wei, Persistent dirac of paths on digraphs and hypergraphs, Foundations of Data Science, 6 (2024), 124-153. doi: 10.3934/fods.2024001.
[36]	J. Townsend, C. P. Micucci, J. H. Hymel, V. Maroulas and K. D. Vogiatzis, Representation of molecular structures with persistent homology for machine learning applications in chemistry, Nature Communications, 11 (2020), 3230. doi: 10.1038/s41467-020-17035-5.
[37]	R. Wang, D. D. Nguyen and G.-W. Wei, Persistent spectral graph, International Journal for Numerical Methods in Biomedical Engineering, 36 (2020), e3376, 27 pp. doi: 10.1002/cnm.3376.
[38]	R. Wang and G.-W. Wei, Persistent path laplacian, Foundations of Data Science, 5 (2023), 26-55. doi: 10.3934/fods.2022015.
[39]	J. Wee, G. Bianconi and K. Xia, Persistent Dirac for molecular representation, Scientific Reports, 13 (2023), 11183. doi: 10.1038/s41598-023-37853-z.
[40]	J. Wee and K. Xia, Forman persistent Ricci curvature (FPRC)-based machine learning models for protein–ligand binding affinity prediction, Briefings in Bioinformatics, 22 (2021), bbab136. doi: 10.1093/bib/bbab136.
[41]	X. Wei and G.-W. Wei, Persistent sheaf laplacians, Foundations of Data Science, 7 (2025), 446-463. doi: 10.3934/fods.2024033.
[42]	X. Wei and G.-W. Wei, Persistent topological Laplacians—a survey, Mathematics, 13 (2025), 208. doi: 10.3390/math13020208.
[43]	K. Xia, D. V. Anand, S. Shikhar and Y. Mu, Persistent homology analysis of osmolyte molecular aggregation and their hydrogen-bonding networks, Physical Chemistry Chemical Physics, 21 (2019), 21038-21048. doi: 10.1039/C9CP03009C.
[44]	K. Xia and G.-W. Wei, Persistent homology analysis of protein structure, flexibility, and folding, International Journal for Numerical Methods in Biomedical Engineering, 30 (2014), 814-844. doi: 10.1002/cnm.2655.
[45]	A. Zomorodian and G. Carlsson, Computing persistent homology, In Proceedings of the Twentieth Annual Symposium on Computational Geometry, 2004,347-356. doi: 10.1145/997817.997870.
[46]	A. Zomorodian and G. Carlsson, Computing persistent homology, Discrete & Computational Geometry, 33 (2005), 249-274. doi: 10.1007/s00454-004-1146-y.