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Sampling-enabled scalable manifold learning unveils the discriminative cluster structure of high-dimensional data

Nature Machine Intelligence (2025)Cite this article

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A preprint version of the article is available at arXiv.

Abstract

As a pivotal branch of machine learning, manifold learning uncovers the intrinsic low-dimensional structure within complex non-linear manifolds in high-dimensional space for visualization, classification, clustering and gaining key insights. Although existing techniques have achieved remarkable successes, they suffer from extensive distortions of cluster structure, which hinders the understanding of underlying patterns. Scalability issues also limit their applicability for handling large-scale data. Here we propose a sampling-based scalable manifold learning technique that enables uniform and discriminative embedding (SUDE) for large-scale and high-dimensional data. It starts by seeking a set of landmarks to construct the low-dimensional skeleton of the entire data and then incorporates the non-landmarks into the learned space by constrained locally linear embedding. We empirically validated the effectiveness of SUDE on synthetic datasets and real-world benchmarks and applied it to analyse single-cell data and detect anomalies in electrocardiogram signals. SUDE exhibits a distinct advantage in scalability with respect to data size and embedding dimension and shows promising performance in cluster separation, integrity and global structure preservation. The experiments also demonstrate notable robustness in embedding quality as the sampling rate decreases.

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Fig. 1: Illustration of SUDE.
Fig. 2: Performance of the PPS algorithm on synthetic datasets.
Fig. 3: Two-dimensional embeddings from five real-world datasets using six manifold learning methods.
Fig. 4: Quantitative comparison of the embedding quality and time efficiency on 13 real-world datasets.
Fig. 5: Performance comparison on scRNA-seq and CyTOF data.
Fig. 6: Recognition of abnormal heartbeats in ECG signals.

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Data availability

The real-world datasets used in this study are available as follows: Wine via University of California Irvine at http://archive.ics.uci.edu/dataset/109/wine, Dermatology via University of California Irvine at http://archive.ics.uci.edu/dataset/33/dermatology, Breast-Cancer via National Taiwan Univeristy at https://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/binary/breast-cancer, Mfeat via University of California Irvine at https://archive.ics.uci.edu/dataset/72/multiple+features, Rice via University of California Irvine at https://archive.ics.uci.edu/dataset/545/rice+cammeo+and+osmancik, Spambase via University of California Irvine at https://archive.ics.uci.edu/dataset/94/spambase, Dry-Bean via University of California Irvine at https://archive.ics.uci.edu/dataset/602/dry+bean+dataset, Shuttle via University of California Irvine at https://archive.ics.uci.edu/dataset/148/statlog+shuttle, CIFAR10 via the University of Toronto at https://www.cs.toronto.edu/~kriz/cifar.html, MNIST via Yann LeCun at https://yann.lecun.com/exdb/mnist/, FMNIST via Github at https://github.com/zalandoresearch/fashion-mnist, AG’s News via the University of Pisa at http://www.di.unipi.it/~gulli/AG_corpus_of_news_articles.html, Yahoo via GitHub at https://github.com/zjulearning/AtSNE, WT via the GEO database with accession no. GSE125708, HRPEAC via the Broad Institute at https://singlecell.broadinstitute.org/single_cell/study/SCP3025/scrna-seq-and-scatac-seq-atlas-of-human-retinal-pigment-epithelium-and-choroid?#study-download), SARS-CoV-2 via the Broad Institute at https://singlecell.broadinstitute.org/single_cell/study/SCP2593/impact-of-variants-and-vaccination-on-nasal-immunity-across-three-waves-of-sars-cov-2#study-download, Levine via FlowRepository with accession no. FR-FCM-ZZPH, QT database via Mathworks at https://www.mathworks.com/supportfiles/SPT/data/QTDatabaseECGData.zip and MIT-BIH long-term database via PhysioNet at https://archive.physionet.org/physiobank/database/ltdb/.

Code availability

The source code for SUDE in MATLAB and Python versions is available via GitHub at https://github.com/ZPGuiGroupWhu/sude (ref. 74) and via Code Ocean at https://doi.org/10.24433/CO.9866909.v3 (ref. 75). The TriMap algorithm is available via OMIQ at http://www.omiq.ai. The Seurat software is available via the Satija Lab at https://satijalab.org/seurat. The TopoAE model is available via GitHub at https://github.com/BorgwardtLab/topological-autoencoders. The P-UMAP model is available via Read the Docs at https://umap-learn.readthedocs.io/en/latest/parametric_umap.html. The Slingshot toolkit is available via GitHub at https://github.com/kstreet13/slingshot.

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Acknowledgements

We thank Y. Tang and Z. Ruan for technical assistance in the analysis of single-cell data. We are also grateful to Y. Liu and R. Wang for language and technical discussion. This paper is supported by the National Natural Science Foundation of China (grant nos. 42090011, 42501573, 42571496, 41971349 and 41930107), Postdoctoral Fellowship Program and China Postdoctoral Science Foundation (grant no. BX20250084), China Postdoctoral Science Foundation (grant no. 2025M770345) and Fundamental Research Funds for the Central Universities (grant nos. 2042024kf0005 and 2042025kf0084).

Author information

Authors and Affiliations

  1. School of Remote Sensing and Information Engineering, Wuhan University, Wuhan, China

    Dehua Peng, Zhipeng Gui & Wenzhang Wei

  2. State Key Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing, Wuhan University, Wuhan, China

    Dehua Peng, Zhipeng Gui, Huayi Wu & Jianya Gong

  3. Collaborative Innovation Center of Geospatial Technology, Wuhan University, Wuhan, China

    Dehua Peng, Zhipeng Gui, Huayi Wu & Jianya Gong

  4. Department of Earth System Science, Stanford University, Stanford, CA, USA

    Fa Li

  5. School of Cyber Science and Engineering, Southeast University, Nanjing, China

    Jie Gui

  6. Purple Mountain Laboratories, Nanjing, China

    Jie Gui

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Contributions

Z.G., H.W. and J. Gong envisioned the study. D.P. and Z.G. designed the algorithm and collected datasets. D.P., W.W. and Z.G. conducted experiments and performed the analysis. D.P., Z.G., W.W. and F.L. wrote the manuscript. H.W. and J. Gui provided advice in analysis.

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Correspondence to Zhipeng Gui.

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Nature Machine Intelligence thanks Dmitry Kobak, Samantha Riesenfeld, Ryan Robinett and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary Notes 1–12, Figs. 1–31 and Tables 1–4.

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Peng, D., Gui, Z., Wei, W. et al. Sampling-enabled scalable manifold learning unveils the discriminative cluster structure of high-dimensional data. Nat Mach Intell (2025). https://doi.org/10.1038/s42256-025-01112-9

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