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Persistent-homology-based machine learning: a survey and a comparative study

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Abstract

A suitable feature representation that can both preserve the data intrinsic information and reduce data complexity and dimensionality is key to the performance of machine learning models. Deeply rooted in algebraic topology, persistent homology (PH) provides a delicate balance between data simplification and intrinsic structure characterization, and has been applied to various areas successfully. However, the combination of PH and machine learning has been hindered greatly by three challenges, namely topological representation of data, PH-based distance measurements or metrics, and PH-based feature representation. With the development of topological data analysis, progresses have been made on all these three problems, but widely scattered in different literatures. In this paper, we provide a systematical review of PH and PH-based supervised and unsupervised models from a computational perspective. Our emphasizes are the recent development of mathematical models and tools, including PH software and PH-based functions, feature representations, kernels, and similarity models. Essentially, this paper can work as a roadmap for the practical application of PH-based machine learning tools. Further, we compare between two types of simplicial complexes (alpha and Vietrois-Rips complexes), two types of feature extractions (barcode statistics and binned features), and three types of machine learning models (support vector machines, tree-based models, and neural networks), and investigate their impacts on the protein secondary structure classification.

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Data availability

The data and codes can be downloaded from https://entuedu-my.sharepoint.com/:f:/g/personal/xiakelin_staff_main_ntu_edu_sg/EvZ-CivdgCdCu90JpIAR3BYBmJwl--DxteRirSvLnAhFHA?e=kBPAP3.

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Funding

This research is partially supported by Nanyang Technological University Startup Grants M4081840 and M4081842, Data Science and Artificial Intelligence Research Centre@NTU M4082115, and Singapore Ministry of Education Academic Research Fund Tier 1 RG109/19, Tier 2 MOE2018-T2-1-033 and MOE-T2EP20120-0013.

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Authors and Affiliations

  1. School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, Singapore

    Chi Seng Pun, Si Xian Lee & Kelin Xia

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  1. Chi Seng Pun
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  2. Si Xian Lee
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  3. Kelin Xia
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Correspondence to Chi Seng Pun or Kelin Xia.

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Appendices

Appendix 1: Properties of the Protein Secondary Structure

This section gives a review of some of the key properties of the secondary structure of proteins. The two main types of protein secondary structure are the alpha (\(\alpha\))-helix and the beta (\(\beta\))-pleated sheets.

The \(\alpha\)-helix has the following properties:

  1. (1)

    Bond length between immediate \(C_{\alpha }\) atom is 3.8Å.

    • This corresponds to the length of typical Betti-0 (Dim-0) bars.

  2. (2)

    Each turn is made up of 3.6 amino acid residues.

    • The formation of Betti-1 (Dim-1) bars can be explained using the slicing technique described in Xia and Wei (2014).

    • The alpha-helix structure is stabalised by the presence of hydrogen bonds between the amine hydrogen N-H and carbonyl C\(=\)O oxygen.

    • Each set of 4 \(C_{\alpha }\) atoms form a one-dimensional loop which contributes to a Betti-1 (Dim-1) bar.

  3. (3)

    Absence of Betti-2 (Dim-2) bars where no cavity is formed since there is insufficient "time" such that the loops are filled up as faces before the cavity can be formed for a single alpha helix.

The \(\beta\)-pleated sheets have the following properties:

  1. (1)

    Bond length/distance between immediate \(C_{\alpha }\) atom in the same strand is also 3.8Å.

    • This corresponds to the length of typical Betti-0 (Dim-0) bars. The bar terminates once the atoms are connected.

  2. (2)

    Each \(\beta\)-pleated sheet is a stretched out polypeptide chain made up of 3 to 10 amino acid residues.

  3. (3)

    The \(\beta\)-pleated sheets are extended structures that are stabalised by hydrogen bonds between residues in adjacent chains.

  4. (4)

    Each strand must be connected to adjacent strands where the shortest distance between \(C_{\alpha }\) and the nearest neighbour in adjacent strand is 4.1Å.

  5. (5)

    Adjacent chains run parallel or antiparallel to one another.

Principal component analysis on binned features

In this appendix, we investigate the effects of principal component analysis (PCA) on PHML for our application in Section 5. We do not involve the tree-based methods with PCA because trees process dimension reduction by their own construction.

There are signs of quite high correlation between adjacent BF as seen in Fig. 6. The use of bins unavoidably suffers from the curse of dimensionality, especially when there are limited number of samples n and \(n\ll p\), where p is the number of features. To tackle such a situation, PCA can be applied to transform features into a few uncorrelated PCs, which can be viewed as new features in a lower dimensional feature space. The downside of such an approach is that the final PCs do not have a clear interpretation to the original bins.

Fig. 6
figure 6

(a) Correlation matrix for the BFs from RC barcodes with 40 bins (where the near-zero variance variables removed). (b) Correlation matrix for the 30 PCs transformed from the same BFs. The blue regions corresponds to high positive correlation and is observed between multiple BFs, in contrast with that in the PCs, where white regions correspond to zero correlation

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In subsequent reports, the experimental results involving principal components transformed from BFs are denoted by an extension “PC". For consistency, only the first 30 PCs will be used for all transformed features using either RC or AC barcodes. The same set of PCs are used as input features for SVMs and (deep) NNs (with dropout). The settings are the same as specified in Sects. 5.2.1 and 5.2.3. Tables 4 and 5 report the corresponding results.

Table 4 Results using SVM with different simplicial complexes and principal components of BFs
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Table 5 Results using NN with different simplicial complexes and principal components of BFs
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By comparing the results in Tables 1 and 4 and Tables 3 and 5, we can see that the use of PCA on BFs does not improve the performance. It implies that the PCA transformation lost information of BFs. In conclusion, it is not recommended that ML algorithms with BFs are incorporated with PCA. However, it does not prohibit PCA from being a powerful visualization tool for the unstructured topological data.

Effects of increasing bin number for binned features

In Tables 6 and 9, the first three or four columns specify the settings of PHML and the remaining columns report the evaluation measurements. The highest overall accuracy number across different bin numbers for a given method is highlighted in red.

Table 6 Results using SVM with different simplicial complexes and BF of different bin numbers
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Table 7 Results using tree-based methods with different simplicial complexes and BF of different bin numbers. The three maxtree numbers in last two rows correspond to the cross-validated number of iterations for different bin numbers
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Table 8 Results using repeated NN with different simplicial complexes and BF of different bin numbers
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Table 9 Results using deep NN with different simplicial complexes and BF of different bin numbers
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Pun, C.S., Lee, S.X. & Xia, K. Persistent-homology-based machine learning: a survey and a comparative study. Artif Intell Rev 55, 5169–5213 (2022). https://doi.org/10.1007/s10462-022-10146-z

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