Manifold Topological Deep Learning for Biomedical Data

2025-02-28

AI Summary

Biomedical images and other data defined on differentiable manifolds pose challenges for conventional deep learning, which typically assumes Euclidean domains. Method: This paper introduces Manifold Topological Deep Learning (MTDL), the first framework to systematically integrate algebraic and differential topology—particularly Hodge theory—into deep learning. Leveraging the Hodge decomposition, MTDL models manifold-valued images as vector fields and enables geometrically aware, orthogonal fusion of gradient, co-gradient, and harmonic components. Integrated with a lightweight CNN architecture, MTDL achieves state-of-the-art performance on MedMNIST v2 (717,287 2D/3D medical images). Contributions: (1) Extends topological deep learning to differentiable manifolds; (2) Establishes a Hodge-theoretic, manifold-structure-aware representation paradigm; (3) Demonstrates superior modeling capacity and generalization on non-Euclidean data.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Representation Learning for Vision
• Humans and AI: Human-in-the-loop Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Recently, topological deep learning (TDL), which integrates algebraic topology with deep neural networks, has achieved tremendous success in processing point-cloud data, emerging as a promising paradigm in data science. However, TDL has not been developed for data on differentiable manifolds, including images, due to the challenges posed by differential topology. We address this challenge by introducing manifold topological deep learning (MTDL) for the first time. To highlight the power of Hodge theory rooted in differential topology, we consider a simple convolutional neural network (CNN) in MTDL. In this novel 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 CNN 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.

Problem

• Extends topological deep learning to differentiable manifolds.
• Introduces Hodge theory for image representation in CNNs.
• Evaluates MTDL on biomedical images, outperforming existing methods.

Innovation

• Integrates Hodge theory with deep learning
• Represents images as smooth manifolds
• Uses CNN for manifold topological deep learning