Manifold topological deep learning for biomedical data

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. The study introduces manifold topological deep learning (MTDL) that extends topological deep learning to data on differentiable manifolds through de Rham-Hodge theory. MTDL outperforms competing methods on various biomedical imaging benchmarks.