June  2022, 4(2): 165-216. doi: 10.3934/fods.2022002

Aspects of topological approaches for data science

1. 

School of Mathematical Sciences, University of Southampton, Southampton, UK

2. 

School of Mathematical Sciences, Center of Topology and Geometry based Technology, Hebei Normal University, Yuhua District, Shijiazhuang, Hebei, 050024 China

3. 

Yanqi Lake Beijing Institute of Mathematica Sciences, Yanqihu, Huairou District, Beijing, 101408 China

4. 

School of Physical and Mathematical Sciences, Nanyang Technological University, SPMS-MAS-05-18, 21 Nanyang Link, 1, Singapore 63737

5. 

Department of Mathematics, Department of Computer Science and Engineering, Department of Biochemistry and Molecular Biology, Michigan State University, MI 48824, USA

*Corresponding author: Jie Wu

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

Received  November 2021 Revised  January 2022 Published  June 2022 Early access  February 2022

Fund Project: 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

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.

Citation: Jelena Grbić, Jie Wu, Kelin Xia, Guo-Wei Wei. Aspects of topological approaches for data science. Foundations of Data Science, 2022, 4 (2) : 165-216. doi: 10.3934/fods.2022002
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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
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
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