Abstract
Recently, topological deep learning (TDL), which integrates algebraic topology with deep neural networks, has achieved significant success in processing point-cloud data and has emerged as a promising paradigm in data science. However, TDL has not been extended to differentiable-manifold data, including images, due to the challenges introduced by differential topology. We address this challenge by introducing a manifold topological deep learning (MTDL) framework. To apply Hodge theory, we integrate it into a streamlined convolutional neural network within the MTDL framework. In this framework, original images are represented as smooth manifolds with vector fields that are decomposed into three orthogonal components based on Hodge theory. These components are then concatenated to form an input image for the convolutional neural network architecture. The performance of MTDL is evaluated using the MedMNIST v2 benchmark database, which comprises 717,287 biomedical images from eleven 2D and six 3D datasets. MTDL significantly outperforms other competing methods, extending TDL to a wide range of data on smooth manifolds.
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Data availability
The 17 MedMNIST datasets used in this study can be found on the MedMNIST official website medmnist.com, and the HAM10000 dataset can be found at the Harvard Dataverse. The DermaMNIST subset is licensed under the Creative Commons Attribution-NonCommercial 4.0 license, and no example images from this dataset are displayed in this article.
Code availability
The code is publicly available via Zenodo at https://doi.org/10.5281/zenodo.1879589444.
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Acknowledgements
This work was supported in part by NIH grants R01GM126189, R01AI164266, and R35GM148196, National Science Foundation grants DMS2052983 and IIS-1900473, Michigan State University Research Foundation, and Bristol-Myers Squibb 65109. The work of YS and GW was supported in part by NIH grants R01EB032716, R01EB031102, and R01HL151561.
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X.L. performed the computational studies, analyzed the data, drafted and revised the manuscript. Z.S. and Y.Y.S. analyzed the data and revised the manuscript. Y.Y.T. and G.W. provided critical review and revisions. G.W.W conceptualized the study, supervised the project, acquired funding, and revised the manuscript.
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Nature Communications thanks Tolga Birdal, who co-reviewed with Yiming Huang; and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.
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Liu, X., Su, Z., Shi, Y. et al. Manifold topological deep learning for biomedical data. Nat Commun (2026). https://doi.org/10.1038/s41467-026-71392-1
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DOI: https://doi.org/10.1038/s41467-026-71392-1
