# Manifold Topological Deep Learning for Biomedical Data

## TL;DR

This paper introduces a new deep learning framework that uses topology to process biomedical images more effectively.

## Contribution

The novel MTDL framework extends topological deep learning to smooth manifolds using Hodge theory for biomedical image analysis.

## Key findings

- MTDL decomposes images into orthogonal vector fields using Hodge theory for CNN input.
- MTDL outperforms existing methods on the MedMNIST v2 benchmark with 717,287 biomedical images.
- The framework successfully processes both 2D and 3D biomedical datasets.

## 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.