Advanced Search
Article Contents
Article Contents

Aspects of topological approaches for data science

  • *Corresponding author: Jie Wu

    *Corresponding author: Jie Wu 

† JG, JW, KX and GW should be considered joint first author

The work of JW was supported in part by Natural Science Foundation of China (NSFC grant no. 11971144) and High-level Scientific Research Foundation of Hebei Province. The third author was supported by Nanyang Technological University Startup Grant M4081842 and Singapore Ministry of Education Academic Research fund Tier 1 RG109/19, Tier 2 MOE-T2EP20220-0010, and Tier 2 MOE-T2EP20120-0013. The work of GWW was supported by NIH grant GM126189, NSF grants DMS-1761320, IIS-1900473, and DMS-2052983, and NASA grant 80NSSC21M0023. We wish to thank the referees most warmly for important suggestions that have improved the exposition of this paper

Abstract Full Text(HTML) Figure(8) / Table(1) Related Papers Cited by
  • We establish a new theory which unifies various aspects of topological approaches for data science, by being applicable both to point cloud data and to graph data, including networks beyond pairwise interactions. We generalize simplicial complexes and hypergraphs to super-hypergraphs and establish super-hypergraph homology as an extension of simplicial homology. Driven by applications, we also introduce super-persistent homology.

    Mathematics Subject Classification: Primary: 55N31; Secondary: 05C65, 05C69, 05C70, 18N50, 55U05, 55U10, 62R40.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Illustration of TDA and TDA-based learning models for data analysis. Generally speaking, all TDA-based learning models have four components, including data, topology, feature and learning. More specifically, data is collected and preprocessed firstly. Second, topological representations and models are constructed to describe the inner structural and interactional information of the data. Note that efficient representations are of key importance to machine learning. Third, a series of topological features are generated by using persistent homology models. Topological-invariant-based features provide a better characterization of the most fundamental and intrinsic properties of the data, thus they have a better generalizability and transferability for machine learning models. Finally, the topological features are combined with machine learning models for various classification and regression tasks

    Figure 2.  Illustration of a super-hypergraph model constructed from the protein-ligand complex (ID: 3E6Y). The ligand (green color) is a drug that is used to cure the disease caused by the protein (red color). The potency and efficacy of the drug is directly determined by the atomic interactions between the ligand and the protein. Traditionally, atomic interactions are modeled by a graph (A). However, graphs can only characterize pair-wise interactions (by edges) and fall short for many-body interactions. Hypergraph models (C) use the hyperedge, i.e., a set of vertices, to represent many-body interactions and have demonstrated great power for biomolecular data analysis (See Section 4 for details). Mathematically, a $ n $-hyperedge contains $ n+1 $ vertices in it. Note that 1-hyperedges are denoted by red ellipses and $ n $-hyperedges ($ n>1 $) are represented by blue ellipses. The super-hypergraph (D) provides an even more flexible representation and incorporates detailed local topology within each hyperedge. Note that the hyperedge in super-hypergraph is a subgraph, i.e., a set of vertices together with edges. If we only consider vertex part of the subgraph, the super-hypergraph reduces to a hypergraph

    Figure 3.  (a) The hypergraph $ {\mathcal{H}} $, where the cross indicates that a vertex is missing. (b) $ \Delta( {\mathcal{H}}) $, the smallest $ \Delta $-set that contains $ {\mathcal{H}} $

    Figure 4.  The hypergraph $ {\mathcal{H}} $ is a standard 2 simplex where the dotted edge is missing

    Figure 5.  Illustration of an element-specific hypergraph model for a protein-ligand complex (ID 3PB3). The binding core region of the complex is decomposed into a series of element-specific atom-sets. The interactions between protein atom-sets and ligand atom-sets are modeled as a series of hypergraphs

    Figure 6.  Illustration of a hypergraph-based filtration process for the protein-ligand complex with ID 3PB3

    Figure 7.  The multi-graph $ G $, which looks the same as the clique complex $ \mathrm{Clique}(G) $

    Figure 8.  (a) The graph $ G $, which is the same as $ \mathrm{Clique}(G) $. (b) The neighborhood complex of $ G $, $ \mathcal{N}(G) $

    Table 1.  opological structures associated to graphs

    Constructions Complex Type Face Type
    clique complex of a simple graph simplicial complex vertex-deletion
    clique complex of a multi-graph $\Delta$-set vertex-deletion
    neighborhood complex simplicial complex vertex-deletion
    Jonsson's graph complex simplicial complex edge-deletion
    path complex of a simple graph hypergraph vertex-deletion
    path complex of a multi-graph/quiver super-hypergraph vertex-deletion
     | Show Table
    DownLoad: CSV
  • [1] Dionysus: The persistent homology software, Software available at http://www.mrzv.org/software/dionysus.
    [2] 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, J. Mach. Learn. Res., 18 (2017), Paper No. 8, 35 pp.
    [3] A. AdcockE. Carlsson and G. Carlsson, The ring of algebraic functions on persistence bar codes, Homology Homotopy Appl., 18 (2016), 381-402.  doi: 10.4310/HHA.2016.v18.n1.a21.
    [4] R. AharoniE. Berger and R. Ziv, Independent systems of representatives in weighted graphs, Combinatorica, 27 (2007), 253-267.  doi: 10.1007/s00493-007-2086-y.
    [5] M. Ahmed, B. T. Fasy and C. Wenk, Local persistent homology based distance between maps, In Proceedings of the 22nd ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, ACM, (2014), 43–52. doi: 10.1145/2666310.2666390.
    [6] R. Anirudh, J. J. Thiagarajan, I. Kim and W. Polonik, Autism spectrum disorder classification using graph kernels on multidimensional time series, preprint, arXiv: 1611.09897.
    [7] E. Babson and D. N. Kozlov, Complexes of graph homomorphisms, Israel J. Math., 152 (2006), 285-312.  doi: 10.1007/BF02771988.
    [8] E. Babson and D. N. Kozlov, Proof of the Lovász conjecture, Ann. of Math., 165 (2007), 965-1007.  doi: 10.4007/annals.2007.165.965.
    [9] W. Bae, J. J. Yoo and J. C. Ye, Beyond deep residual learning for image restoration: Persistent homology-guided manifold simplification, In IEEE Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), (2017), 1141–1149. doi: 10.1109/CVPRW.2017.152.
    [10] R. Balakrishnan and K. Ranganathan, A Textbook of Graph Theory, 2$^{nd}$ edition, Universitext, Springer, New York, 2012. doi: 10.1007/978-1-4614-4529-6.
    [11] J. A. Barmak, Star clusters in independence complexes of graphs, Adv. Math., 241 (2013), 33-57.  doi: 10.1016/j.aim.2013.03.016.
    [12] F. BattistonG. CencettiI. IacopiniV. LatoraM. LucasA. PataniaJ.-G. Young and G. Petri, Networks beyond pairwise interactions: Structure and dynamics, Phys. Rep., 874 (2020), 1-92.  doi: 10.1016/j.physrep.2020.05.004.
    [13] U. Bauer, Ripser: A lean C++ code for the computation of Vietoris-Rips persistence barcodes, Software available at https://github. com/Ripser/ripser.
    [14] U. Bauer, M. Kerber and J. Reininghaus, Distributed computation of persistent homology, In Meeting on Algorithm Engineering and Experiments (ALENEX), SIAM, (2014), 31–38. doi: 10.1137/1.9781611973198.4.
    [15] U. BauerM. KerberJ. Reininghaus and H. Wagner, PHAT–persistent homology algorithms toolbox, Mathematical software¡aICMS, 8592 (2014), 137-143.  doi: 10.1007/978-3-662-44199-2_24.
    [16] P. Bendich, D. Cohen-Steiner, H. Edelsbrunner, J. Harer and D. Morozov, Inferring local homology from sampled stratified spaces, In IEEE Symposium on Foundations of Computer Science (FOCS'07), (2007), 536–546. doi: 10.1109/FOCS.2007.45.
    [17] P. BendichH. Edelsbrunner and M. Kerber, Computing robustness and persistence for images, IEEE Transactions on Visualization and Computer Graphics, 16 (2010), 1251-1260.  doi: 10.1109/TVCG.2010.139.
    [18] P. Bendich, E. Gasparovic, J. Harer, R. Izmailov and L. Ness, Multi-scale local shape analysis and feature selection in machine learning applications, In International Joint Conference on Neural Networks (IJCNN), IEEE, (2015), 1–8. doi: 10.1109/IJCNN.2015.7280428.
    [19] P. Bendich, B. Wang and S. Mukherjee, Local homology transfer and stratification learning, In Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, (2012), 1355–1370.
    [20] M. G. BergomiM. FerriP. Vertechi and L. Zuffi, Beyond topological persistence: Starting from networks, Mathematics, 9 (2021).  doi: 10.3390/math9233079.
    [21] J. BinchiE. MerelliM. RuccoG. Petri and F. Vaccarino, jholes: A tool for understanding biological complex networks via clique weight rank persistent homology, Electron. Notes Theor. Comput. Sci., 306 (2014), 5-18.  doi: 10.1016/j.entcs.2014.06.011.
    [22] T. BonisM. OvsjanikovS. Oudot and F. Chazal, Persistence-based pooling for shape pose recognition, Computational Topology in Image Context, 9667 (2016), 19-29.  doi: 10.1007/978-3-319-39441-1_3.
    [23] S. BressanJ. LiS. Ren and J. Wu, The embedded homology of hypergraphs and applications, Asian J. Math., 23 (2019), 479-500.  doi: 10.4310/AJM.2019.v23.n3.a6.
    [24] P. Bubenik, Statistical topological data analysis using persistence landscapes, J. Mach. Learn. Res., 16 (2015), 77-102. 
    [25] P. Bubenik and P. T. Kim, A statistical approach to persistent homology, Homology Homotopy Appl., 9 (2007), 337-362.  doi: 10.4310/HHA.2007.v9.n2.a12.
    [26] P. Bubenik and T. Vergili, Topological spaces of persistence modules and their properties, J. Appl. Comput. Topol., 2 (2018), 233-269.  doi: 10.1007/s41468-018-0022-4.
    [27] P. J. Cameron, Automorphisms and cohomology of switching classes, J. Combinatorial Theory Ser. B, 22 (1977), 297-298.  doi: 10.1016/0095-8956(77)90079-X.
    [28] P. J. Cameron, Cohomological aspects of two-graphs, Math. Z., 157 (1977), 101-119.  doi: 10.1007/BF01215145.
    [29] Z. CangL. Mu and G.-W. Wei, Representability of algebraic topology for biomolecules in machine learning based scoring and virtual screening, PLOS Computational Biology, 14 (2018), 1-44.  doi: 10.1371/journal.pcbi.1005929.
    [30] Z. Cang and G. Wei, Integration of element specific persistent homology and machine learning for protein-ligand binding affinity prediction, International Journal for Numerical Methods in Biomedical Engineering, 34 (2018), e2914.  doi: 10.1002/cnm.2914.
    [31] Z. Cang and G.-W. Wei, Analysis and prediction of protein folding energy changes upon mutation by element specific persistent homology, Bioinformatics, 33 (2017), 3549-3557.  doi: 10.1093/bioinformatics/btx460.
    [32] 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), 1-27.  doi: 10.1371/journal.pcbi.1005690.
    [33] G. CarlssonT. IshkhanovV. Silva and A. Zomorodian, On the local behavior of spaces of natural images, Int. J. Comput. Vis., 76 (2008), 1-12.  doi: 10.1007/s11263-007-0056-x.
    [34] G. CarlssonG. Singh and A. Zomorodian, Computing multidimensional persistence, Algorithms and Computation, 5878 (2009), 730-739.  doi: 10.1007/978-3-642-10631-6_74.
    [35] G. Carlsson and A. Zomorodian, The theory of multidimensional persistence, Discrete Comput. Geom., 42 (2009), 71-93.  doi: 10.1007/s00454-009-9176-0.
    [36] G. Carlsson, Topology and data, Bull. Amer. Math. Soc. (N.S.), 46 (2009), 255-308.  doi: 10.1090/S0273-0979-09-01249-X.
    [37] A. Cerri and C. Landi, The persistence space in multidimensional persistent homology, Discrete Geometry for Computer Imagery, 7749 (2013), 180-191.  doi: 10.1007/978-3-642-37067-0_16.
    [38] F. Chazal, D. Cohen-Steiner, M. Glisse, L. J. Guibas and S. Y. Oudot, Proximity of persistence modules and their diagrams, In SCG '09: Proceedings of the twenty-fifth annual symposium on Computational Geometry, (2009), 237–246. doi: 10.1145/1542362.1542407.
    [39] F. Chazal, V. de Silva, M. Glisse and S. Oudot, The Structure and Stability of Persistence Modules, SpringerBriefs in Mathematics, Springer, [Cham], 2016. doi: 10.1007/978-3-319-42545-0.
    [40] F. Chazal, B. Fasy, F. Lecci, B. Michel, A. Rinaldo and L. Wasserman, Subsampling methods for persistent homology, In Proceedings of the 32nd International Conference on Machine Learning, (eds. F. Bach and D. Blei), PMLR, Lille, France, 37 (2015), 2143–2151.
    [41] F. Chazal and B. Michel, An introduction to topological data analysis: Fundamental and practical aspects for data scientists, Front. Artif. Intell, 2021. doi: 10.3389/frai.2021.667963.
    [42] Y. Cheng and A. L. Wells Jr., Switching classes of directed graphs, J. Combin. Theory Ser. B, 40 (1986), 169-186.  doi: 10.1016/0095-8956(86)90075-4.
    [43] I. ChevyrevV. Nanda and H. Oberhauser, Persistence paths and signature features in topological data analysis, IEEE Transactions on Pattern Analysis and Machine Intelligence, 42 (2018), 192-202.  doi: 10.1109/TPAMI.2018.2885516.
    [44] F. R. K. Chung and R. L. Graham, Cohomological aspects of hypergraphs, Trans. Amer. Math. Soc., 334 (1992), 365-388.  doi: 10.1090/S0002-9947-1992-1089416-0.
    [45] D. Cohen-Steiner, H. Edelsbrunner and D. Morozov, Vines and vineyards by updating persistence in linear time, In Computational geometry (SCG'06), (2006), 119–126. doi: 10.1145/1137856.1137877.
    [46] W. Crawley-Boevey, Decomposition of pointwise finite-dimensional persistence modules, J. Algebra Appl., 14 (2015), 1550066, 8 pp. doi: 10.1142/S0219498815500668.
    [47] E. B. Curtis, Simplicial homotopy theory, Advances in Math., 6 (1971), 107-209.  doi: 10.1016/0001-8708(71)90015-6.
    [48] V. de Silva and R. Ghrist, Homological sensor networks, Notices Amer. Math. Soc., 54 (2007), 10-17. 
    [49] V. De SilvaD. Morozov and M. Vejdemo-Johansson, Persistent cohomology and circular coordinates, Discrete Comput. Geom., 45 (2011), 737-759.  doi: 10.1007/s00454-011-9344-x.
    [50] T. K. Dey and S. Mandal, Protein classification with improved topological data analysis, In LIPIcs. Leibniz Int. Proc. Inform., 113 (2018), 13 pp.
    [51] R. Diestel, Graph Theory, vol. 173 of Graduate Texts in Mathematics, 5$^{th}$ edition, Graduate Texts in Mathematics, 173. Springer, Berlin, 2017. doi: 10.1007/978-3-662-53622-3.
    [52] A. Dimakis and F. Müller-Hoissen, Differential calculus and gauge theory on finite sets, J. Phys. A, 27 (1994), 3159-3178.  doi: 10.1088/0305-4470/27/9/028.
    [53] A. Dimakis and F. Müller-Hoissen, Discrete differential calculus: Graphs, topologies, and gauge theory, J. Math. Phys., 35 (1994), 6703-6735.  doi: 10.1063/1.530638.
    [54] A. Dochtermann, Hom complexes and homotopy theory in the category of graphs, European J. Combin., 30 (2009), 490-509.  doi: 10.1016/j.ejc.2008.04.009.
    [55] A. M. Duval and V. Reiner, Shifted simplicial complexes are Laplacian integral, Trans. Amer. Math. Soc., 354 (2002), 4313-4344.  doi: 10.1090/S0002-9947-02-03082-9.
    [56] B. Dwork, On the zeta function of a hypersurface, Inst. Hautes Études Sci. Publ. Math., (1962), 5-68.  doi: 10.1007/BF02684275.
    [57] H. EdelsbrunnerD. Letscher and A. Zomorodian, Topological persistence and simplification, Discrete Comput. Geom., 28 (2002), 511-533.  doi: 10.1007/s00454-002-2885-2.
    [58] R. Ehrenborg and G. Hetyei, The topology of the independence complex, European J. Combin., 27 (2006), 906-923.  doi: 10.1016/j.ejc.2005.04.010.
    [59] E. Emtander, Betti numbers of hypergraphs, Comm. Algebra, 37 (2009), 1545-1571.  doi: 10.1080/00927870802098158.
    [60] A. Engström, Independence complexes of claw-free graphs, European J. Combin., 29 (2008), 234-241.  doi: 10.1016/j.ejc.2006.09.007.
    [61] B. T. Fasy, J. Kim, F. Lecci and C. Maria, Introduction to the r package tda, preprint, arXiv: 1411.1830.
    [62] B. T. Fasy and B. Wang, Exploring persistent local homology in topological data analysis, In IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), (2016), 6430–6434. doi: 10.1109/ICASSP.2016.7472915.
    [63] R. Forman, Morse theory for cell complexes, Adv. Math., 134 (1998), 90-145.  doi: 10.1006/aima.1997.1650.
    [64] P. Frosini and C. Landi, Persistent Betti numbers for a noise tolerant shape-based approach to image retrieval, Computer Analysis of Images and Patterns, 6854 (2011), 294-301.  doi: 10.1007/978-3-642-23672-3_36.
    [65] P. Gabriel, Unzerlegbare Darstellungen. I, Manuscripta Math., 6 (1972), 71-103.  doi: 10.1007/BF01298413.
    [66] R. Ghrist, Barcodes: The persistent topology of data, Bull. Amer. Math. Soc., 45 (2008), 61-75.  doi: 10.1090/S0273-0979-07-01191-3.
    [67] N. Giansiracusa, R. Giansiracusa and C. Moon, Persistent homology machine learning for fingerprint classification, IEEE International Conference On Machine Learning And Applications (ICMLA), 2019. doi: 10.1109/ICMLA.2019.00201.
    [68] A. Grigorian, Y. Lin, Y. Muranov and S.-T. Yau, Homologies of path complexes and digraphs, arXiv.
    [69] A. GrigorianY. Muranov and S.-T. Yau, Homologies of digraphs and künneth formulas, Comm. Anal. Geom., 25 (2017), 969-1018.  doi: 10.4310/CAG.2017.v25.n5.a4.
    [70] A. Grigor'yanY. V. Muranov and S.-T. Yau, Graphs associated with simplicial complexes, Homology Homotopy Appl., 16 (2014), 295-311.  doi: 10.4310/HHA.2014.v16.n1.a16.
    [71] A. Grigor'yan and Y. V. Muranov, Cohomology theories of simplicial complexes, algebras, and digraphs.,
    [72] A. A. Grigor'yanĬ. LinY. V. Muranov and S. Yau, Path complexes and their homologies, Fundam. Prikl. Mat., 21 (2016), 79-128. 
    [73] A. Grigor'yanR. JimenezY. Muranov and S.-T. Yau, On the path homology theory of digraphs and Eilenberg-Steenrod axioms, Homology Homotopy Appl., 20 (2018), 179-205.  doi: 10.4310/HHA.2018.v20.n2.a9.
    [74] A. Grigor'yan, R. Jimenez, Y. Muranov and S.-T. Yau, Homology of path complexes and hypergraphs, Topology Appl., 267 (2019), 106877, 25 pp. doi: 10.1016/j.topol.2019.106877.
    [75] A. Grigor'yanY. LinY. Muranov and S.-T. Yau, Homotopy theory for digraphs, Pure Appl. Math. Q., 10 (2014), 619-674.  doi: 10.4310/PAMQ.2014.v10.n4.a2.
    [76] A. Grigor'yanY. LinY. Muranov and S.-T. Yau, Cohomology of digraphs and (undirected) graphs, Asian J. Math., 19 (2015), 887-931.  doi: 10.4310/AJM.2015.v19.n5.a5.
    [77] A. Grigor'yanY. MuranovV. Vershinin and S.-T. Yau, Path homology theory of multigraphs and quivers, Forum Math., 30 (2018), 1319-1337.  doi: 10.1515/forum-2018-0015.
    [78] W. GuoK. ManoharS. L. Brunton and A. G. Banerjee, Sparse-tda: Sparse realization of topological data analysis for multi-way classification, IEEE Transactions on Knowledge and Data Engineering, 30 (2018), 1403-1408.  doi: 10.1109/TKDE.2018.2790386.
    [79] Y. S. Han, J. Yoo and J. C. Ye, Deep residual learning for compressed sensing ct reconstruction via persistent homology analysis, preprint, arXiv: 1611.06391.
    [80] A. HatcherAlgebraic Topology, Cambridge University Press, Cambridge, 2002. 
    [81] Y. HiraokaT. NakamuraA. HirataE. G. EscolarK. Matsue and Y. Nishiura, Hierarchical structures of amorphous solids characterized by persistent homology, Proceedings of the National Academy of Sciences, 113 (2016), 7035-7040.  doi: 10.1073/pnas.1520877113.
    [82] D. Horak and J. Jost, Spectra of combinatorial Laplace operators on simplicial complexes, Adv. Math., 244 (2013), 303-336.  doi: 10.1016/j.aim.2013.05.007.
    [83] I. M. JamesFibrewise Topology, Cambridge Tracts in Mathematics, 91. Cambridge University Press, Cambridge, 1989.  doi: 10.1017/CBO9780511896835.
    [84] J. Jonsson, Simplicial Complexes of Graphs, Lecture Notes in Mathematics, 1928. Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-75859-4.
    [85] S. Kališnik, Tropical coordinates on the space of persistence barcodes, Foundations of Computational Mathematics, 1–29.
    [86] M. Kontsevich, Derived Grothendieck-Teichmüller group and graph complexes [after T. Willwacher], Exposés, 1126 (2019), 183-211.  doi: 10.24033/ast.
    [87] D. N. Kozlov, Complexes of directed trees, J. Combin. Theory Ser. A, 88 (1999), 112-122.  doi: 10.1006/jcta.1999.2984.
    [88] D. N. Kozlov, Discrete Morse theory for free chain complexes, C. R. Math., 340 (2005), 867-872.  doi: 10.1016/j.crma.2005.04.036.
    [89] D. N. Kozlov, Simple homotopy types of Hom-complexes, neighborhood complexes, Lovász complexes, and atom crosscut complexes, Topology Appl., 153 (2006), 2445-2454.  doi: 10.1016/j.topol.2005.09.005.
    [90] M. KramárR. LevangerJ. TithofB. SuriM. XuM. PaulM. F. Schatz and K. Mischaikow, Analysis of Kolmogorov flow and Rayleigh-Bénard convection using persistent homology, Phys. D, 334 (2016), 82-98.  doi: 10.1016/j.physd.2016.02.003.
    [91] M. KramárA. GoulletL. Kondic and K. Mischaikow, Persistence of force networks in compressed granular media, Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics, 87 (2013), 042207. 
    [92] C. Li, M. Ovsjanikov and F. Chazal, Persistence-based structural recognition, In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (2014), 1995–2002. doi: 10.1109/CVPR.2014.257.
    [93] X. LiuX. WangJ. Wu and K. Xia, Hypergraph-based persistent cohomology (HPC) for molecular representations in drug design, Briefings in Bioinformatics, 22 (2021).  doi: 10.1093/bib/bbaa411.
    [94] L. Lovász, Kneser's conjecture, chromatic number, and homotopy, J. Combin. Theory Ser. A, 25 (1978), 319-324.  doi: 10.1016/0097-3165(78)90022-5.
    [95] N. MakarenkoM. KalimoldayevI. Pak and A. Yessenaliyeva, Texture recognition by the methods of topological data analysis, Open Engineering, 6 (2016).  doi: 10.1515/eng-2016-0044.
    [96] C. L. Mallows and N. J. A. Sloane, Two-graphs, switching classes and Euler graphs are equal in number, SIAM J. Appl. Math., 28 (1975), 876-880.  doi: 10.1137/0128070.
    [97] C. Maria, Filtered complexes, in GUDHI User and Reference Manual, GUDHI Editorial Board, 2015, https://gudhi.inria.fr/doc/3.4.1/group__simplex__tree.html.
    [98] Z. Meng and K. Xia, Persistent spectral–based machine learning (perspect ml) for protein-ligand binding affinity prediction, Science Advances, 7 (2021), eabc5329.  doi: 10.1126/sciadv.abc5329.
    [99] W. Mielants and H. Leemans, $Z_{2}$-cohomology of projective spaces of odd order, In Combinatorics '81 (Rome, 1981), Ann. Discrete Math., North-Holland, Amsterdam-New York, 18 (1983), 635–651.
    [100] K. MischaikowM. MrozekJ. Reiss and A. Szymczak, Construction of symbolic dynamics from experimental time series, Physical Review Letters, 82 (1999), 1144-1147.  doi: 10.1103/PhysRevLett.82.1144.
    [101] K. Mischaikow and V. Nanda, Morse theory for filtrations and efficient computation of persistent homology, Discrete Comput. Geom., 50 (2013), 330-353.  doi: 10.1007/s00454-013-9529-6.
    [102] J. R. Munkres, Elements of Algebraic Topology, Addison-Wesley Publishing Company, Menlo Park, CA, 1984.
    [103] T. NakamuraY. HiraokaA. HirataE. G. Escolar and Y. Nishiura, Persistent homology and many-body atomic structure for medium-range order in the glass, Nanotechnology, 26 (2015), 304001.  doi: 10.1088/0957-4484/26/30/304001.
    [104] V. Nanda, Perseus: The persistent homology software, Software available at http://www.sas.upenn.edu/ vnanda/perseus.
    [105] D. D. Nguyen, Z. X. Cang and G. W. Wei, A review of mathematical representations of biomolecular data, Physical Chemistry Chemical Physics, 2020. doi: 10.1039/C9CP06554G.
    [106] D. NguyenK. GaoM. Wang and G.-W. Wei, MathDL: Mathematical deep learning for D3R grand challenge 4, Journal of Computer-Aided Molecular Design, 34 (2020), 131-147.  doi: 10.1007/s10822-019-00237-5.
    [107] P. NiyogiS. Smale and S. Weinberger, A topological view of unsupervised learning from noisy data, SIAM J. Comput., 40 (2011), 646-663.  doi: 10.1137/090762932.
    [108] I. ObayashiY. Hiraoka and M. Kimura, Persistence diagrams with linear machine learning models, J. Appl. Comput. Topol., 1 (2018), 421-449.  doi: 10.1007/s41468-018-0013-5.
    [109] D. PachauriC. HinrichsM. K. ChungS. C. Johnson and V. Singh, Topology-based kernels with application to inference problems in alzheimer's disease, IEEE Transactions on Medical Imaging, 30 (2011), 1760-1770.  doi: 10.1109/TMI.2011.2147327.
    [110] A. Parks, S. Lipscomb and N. S. W. C. D. VA., Homology and Hypergraph Acyclicity: A Combinatorial Invariant For Hypergraphs, Defense Technical Information Center, 1991, https://apps.dtic.mil/sti/citations/ADA241584.
    [111] F. T. Pokorny, C. H. Ek, H. Kjellström and D. Kragic, Persistent homology for learning densities with bounded support, In Proceedings of the 25th International Conference on Neural Information Processing Systems, NIPS'12, Curran Associates Inc., Red Hook, NY, USA, 2 (2012), 1817–1825.
    [112] T. QaiserY. W. TsangD. TaniyamaN. SakamotoK. NakaneD. Epstein and N. Rajpoot, Fast and accurate tumor segmentation of histology images using persistent homology and deep convolutional features, Medical Image Analysis, 55 (2019), 1-14.  doi: 10.1016/j.media.2019.03.014.
    [113] M. S. Rahman, Basic Graph Theory, Undergraduate Topics in Computer Science, Springer, Cham, 2017. doi: 10.1007/978-3-319-49475-3.
    [114] G. Rebala, A. Ravi and S. Churiwala, An Introduction to Machine Learning, 2019. doi: 10.1007/978-3-030-15729-6.
    [115] J. Reininghaus, D. Günther, I. Hotz, S. Prohaska and H.-C. Hege, TADD: A computational framework for data analysis using discrete Morse theory, In Mathematical Software—ICMS 2010, Lecture Notes in Comput. Sci., 6327 (2010), 198–208.
    [116] S. RenC. Wu and J. Wu, Weighted persistent homology, Rocky Mountain J. Math., 48 (2018), 2661-2687.  doi: 10.1216/rmj-2018-48-8-2661.
    [117] V. Robins and K. Turner, Principal component analysis of persistent homology rank functions with case studies of spatial point patterns, sphere packing and colloids, Phys. D, 334 (2016), 99-117.  doi: 10.1016/j.physd.2016.03.007.
    [118] V. Robins, Computational topology for point data: Betti numbers of $\alpha$-shapes, Morphology of Condensed Matter, 600 (2002), 261-274.  doi: 10.1007/3-540-45782-8_11.
    [119] M. SaadatfarH. TakeuchiV. RobinsN. Francois and Y. Hiraoka, Pore configuration landscape of granular crystallization, Nature communications, 8 (2017), 15082.  doi: 10.1038/ncomms15082.
    [120] A. Said and V. Torra, Data Science in Practice, Studies in Big Data, Springer International Publishing, 2019. doi: 10.1007/978-3-319-97556-6.
    [121] R. Schiffler, Quiver Representations, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, Cham, 2014. doi: 10.1007/978-3-319-09204-1.
    [122] J. J. Seidel, A survey of two-graphs, In Colloquio Internazionale Sulle Teorie Combinatorie (Rome, 1973), Tomo I, Atti dei Convegni Lincei, (1976), 481–511.
    [123] J. J. Seidel and D. E. Taylor, Two-graphs, a second survey, In Algebraic Methods in Graph Theory, Vol. I, II (Szeged, 1978), Colloq. Math. Soc. János Bolyai, 25 (1981), 689–711.
    [124] L. M. Seversky, S. Davis and M. Berger, On time-series topological data analysis: New data and opportunities, In IEEE Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), (2016), 1014–1022.
    [125] G. Singh, F. Memoli, T. Ishkhanov, G. Sapiro, G. Carlsson and D. L. Ringach, Topological analysis of population activity in visual cortex, Journal of Vision, 8 (2008). doi: 10.1167/8.8.11.
    [126] P. Skraba, M. Ovsjanikov, F. Chazal and L. Guibas, Persistence-based segmentation of deformable shapes, In IEEE Computer Society Conference on Computer Vision and Pattern Recognition - Workshops, (2010), 45–52.
    [127] A. Tausz, M. Vejdemo-Johansson and H. Adams, Javaplex: A research software package for persistent (co)homology, Software available at http://code.google.com/p/javaplex, 2011.
    [128] J. Tierny, Topological Data Analysis for Scientific Visualization, Springer-Verlag, Berlin, 2017. doi: 10.1007/978-3-319-71507-0.
    [129] K. TurnerS. Mukherjee and D. M. Boyer, Persistent homology transform for modeling shapes and surfaces, Inf. Inference, 3 (2014), 310-344.  doi: 10.1093/imaiai/iau011.
    [130] Y. Umeda, Time series classification via topological data analysis, Transactions of the Japanese Society for Artificial Intelligence, 32 (2017), 1-12.  doi: 10.1527/tjsai.D-G72.
    [131] M. WangZ. Cang and G.-W. Wei, A topology-based network tree for the prediction of protein-protein binding affinity changes following mutation, Nature Machine Intelligence, 2 (2020), 116-123.  doi: 10.1038/s42256-020-0149-6.
    [132] R. Wang, D. D. Nguyen and G.-W. Wei, Persistent spectral graph, Int. J. Numer. Methods Biomed. Eng., 36 (2020), e3376, 27 pp. doi: 10.1002/cnm.3376.
    [133] Y. Wang, H. Ombao and M. K. Chung et al., Persistence landscape of functional signal and its application to epileptic electroencaphalogram data, ENAR Distinguished Student Paper Award.
    [134] 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).  doi: 10.1093/bib/bbab136.
    [135] J. Wee and K. Xia, Ollivier persistent ricci curvature-based machine learning for the protein–ligand binding affinity prediction, J. Chem. Inf. Model., 61 (2021), 1617-1626.  doi: 10.1021/acs.jcim.0c01415.
    [136] G.-W. Wei, Persistent homology analysis of biomolecular data, SIAM NEWS, 2017, https://sinews.siam.org/Details-Page/persistent-homology-analysis-of-biomolecular-data.
    [137] G. WeiD. Nguyen and Z. Cang, System and methods for machine learning for drug design and discovery, US Patent App., 16 (2019), 239-327. 
    [138] A. L. Wells Jr., Even signings, signed switching classes, and $(-1, 1)$-matrices, J. Combin. Theory Ser. B, 36 (1984), 194-212.  doi: 10.1016/0095-8956(84)90025-X.
    [139] J. Wu, Simplicial objects and homotopy groups, In Braids, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., World Sci. Publ., Hackensack, NJ, 19 (2010), 31–181. doi: 10.1142/9789814291415_0002.
    [140] K. Xia and G.-W. Wei, Persistent homology analysis of protein structure, flexibility, and folding, Int. J. Numer. Methods Biomed. Eng., 30 (2014), 814-844.  doi: 10.1002/cnm.2655.
    [141] N. Yadav, A. Yadav and M. Kumar et al., An Introduction to Neural Network Methods for Differential Equations, SpringerBriefs in Applied Sciences and Technology. Springer, Dordrecht, 2015. doi: 10.1007/978-94-017-9816-7.
    [142] T. Zaslavsky, Characterizations of signed graphs, J. Graph Theory, 5 (1981), 401-406.  doi: 10.1002/jgt.3190050409.
    [143] M. ZeppelzauerB. ZielińskiM. Juda and M. Seidl, A study on topological descriptors for the analysis of 3d surface texture, Computer Vision and Image Understanding, 167 (2018), 74-88.  doi: 10.1016/j.cviu.2017.10.012.
    [144] Z. F. Zhang, Y. Song, H. C. Cui, J. Wu, F. Schwartz and H. R. Qi, Early mastitis diagnosis through topological analysis of biosignals from low-voltage alternate current electrokinetics, In Engineering in Medicine and Biology Society (EMBC), 2015 37th Annual International Conference of the IEEE, (2015), 542–545. doi: 10.1109/EMBC.2015.7318419.
    [145] Z. ZhouY. Z. HuangL. Wang and T. N. Tan, Exploring generalized shape analysis by topological representations, Pattern Recognition Letters, 87 (2017), 177-185.  doi: 10.1016/j.patrec.2016.04.002.
    [146] X. J. Zhu, Persistent homology: An introduction and a new text representation for natural language processing., In IJCAI, (2013), 1953–1959.
    [147] B. ZielinskiM. Juda and M. Zeppelzauer, Persistence codebooks for topological data analysis, Artificial Intelligence Review volume, 54 (2021), 1969-2009.  doi: 10.1007/s10462-020-09897-4.
    [148] A. Zomorodian and G. Carlsson, Computing persistent homology, Discrete Comput. Geom., 33 (2005), 249-274.  doi: 10.1007/s00454-004-1146-y.
  • 加载中




Article Metrics

HTML views(504) PDF downloads(508) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint