Combinatorial Network-Based Manifold Topological Deep Learning for Image Analysis

Alice Wachira; Xiang Liu; Zhe Su; Yiying Tong; Ge Wang; Guo-Wei Wei

cs.CVSep 21, 2026

Combinatorial Network-Based Manifold Topological Deep Learning for Image Analysis

Authors: Alice Wachira, Xiang Liu, Zhe Su, Yiying Tong, Ge Wang, Guo-Wei Wei

Organizations: Department of Mathematics, University of Georgia, Athens, GA 30602, USA. · Department of Mathematics and Statistics, Auburn University, AL, 36849, USA. · Department of Computer Science and Engineering, Michigan State University, MI 48824, USA. · Biomedical Imaging Center, Rensselaer Polytechnic Institute, NY 12180, USA. ·

Abstract

Medical image analysis remains fundamentally challenging because of the intricate geometric and topological structures present in medical data. Conventional convolutional neural networks model images as regular Euclidean grids, limiting their ability to preserve geometric relationships and higher-order structural information. Recently, manifold topological deep learning (MTDL) has emerged as a promising paradigm that integrates deep learning with geometric and topological representations. Nevertheless, existing methods have not yet fully exploited discrete manifold structures within combinatorial complex neural networks. To bridge this gap, we introduce CNMTDL, a MTDL framework that integrates Hodge decomposition with a combinatorial attention mechanism. In our approach, medical images are represented as discrete manifolds and decomposed into three Hodge components. Features extracted from these components are concatenated and embedded into a combinatorial complex architecture, enabling enhanced higher-order message passing between $0$ -cells and $2$ -cells through attention-based blocks. We evaluate CNMTDL on six two-dimensional and three-dimensional datasets from the MedMNIST v2 benchmark, demonstrating its effectiveness for medical image analysis.

Figures & tables

Figure 1: Workflow of the CNMTDL architecture. (A) The input image is modeled as a discrete manifold on a Cartesian grid with prescribed normal and tangential boundary conditions. (B) A vector field is constructed on the manifold representation. (C) The discrete Hodge Laplacian separates the vector field into harmonic and non-harmonic spectral components. (D) Hodge decomposition yields curl-free, divergence-free and harmonic components which are concatenated to form a new image representation (E). (F) The resulting multi-channel representation is passed to a CCNN to perform the image classification (G).

Models	Retina		Derma		Blood		OrganC		OrganS		OrganA
Models	AUC	ACC	AUC	ACC	AUC	ACC	AUC	ACC	AUC	ACC	AUC	ACC
ResNet-18 (28) [ 26 ]	0.717	0.524	0.917	0.735	0.998	0.958	0.992	0.900	0.972	0.782	0.997	0.935
ResNet-50 (28) [ 26 ]	0.726	0.528	0.913	0.735	0.997	0.956	0.992	0.905	0.972	0.770	0.997	0.935
ResNet-18 (224) [ 26 ]	0.710	0.493	0.920	0.754	0.998	0.963	0.994	0.920	0.974	0.778	0.998	0.951
ResNet-50 (224) [ 26 ]	0.716	0.511	0.912	0.731	0.997	0.950	0.993	0.911	0.975	0.785	0.998	0.947
auto-sklearn [ 27 ]	0.690	0.515	0.902	0.719	0.984	0.878	0.976	0.829	0.945	0.672	0.963	0.762

Table 1: Comparison of our proposed model with previous benchmark models on selected MedMNIST v2 2D datasets using ACC and AUC. Baseline results are derived from previous studies [ 5 , 19 , 20 , 21 , 22 , 23 , 24 , 25 ] . Best results are presented in bold.

Models	Fracture		Adrenal		Vessel		Organ		Nodule		Synapse
Models	AUC	ACC	AUC	ACC	AUC	ACC	AUC	ACC	AUC	ACC	AUC	ACC
ResNet-18 [ 26 ] + 2.5D [ 34 ]	0.587	0.451	0.718	0.772	0.748	0.846	0.977	0.788	0.838	0.835	0.634	0.696
ResNet-18 [ 26 ] + 3D [ 34 ]	0.712	0.508	0.827	0.721	0.874	0.877	0.996	0.907	0.863	0.844	0.820	0.745
ResNet-18 [ 26 ] + ACS [ 34 ]	0.714	0.497	0.839	0.754	0.930	0.928	0.994	0.900	0.873	0.847	0.705	0.722
ResNet-50 [ 26 ] + 2.5D [ 34 ]	0.552	0.397	0.732	0.763	0.751	0.877	0.974	0.769	0.835	0.848	0.669	0.735
ResNet-50 [ 26 ] + 3D [ 34 ]	0.725	0.494	0.828	0.745	0.907	0.918	0.994	0.883	0.875	0.847	0.851	0.795

Table 2: Comparison of our proposed model with previous benchmark models on selected MedMNIST v2 3D datasets using ACC and AUC. Baseline results are derived from previous studies [ 5 , 32 , 25 , 24 , 33 ] . Best results are presented in bold.

Figure 2: CNMTDL performance and embedding separability on MedMNIST 2D datasets. UMAP visualization of ground truth (A) and predicted class labels (B). (C) Average AUC and ACC across six MedMNIST v2 2D datasets for CNMTDL and benchmark models.

Figure 3: CNMTDL performance on MedMNIST v2 3D datasets. UMAP embeddings colored by ground truth (A) and predicted class labels (B). (C) Average AUC and ACC across six MedMNIST 3D datasets for CNMTDL and benchmark models.

Models	Retina		Blood		OrganC		OrganS		Vessel		Adrenal		Synapse
Models	AUC	ACC	AUC	ACC	AUC	ACC	AUC	ACC	AUC	ACC	AUC	ACC	AUC	ACC
$0$ -cell CCNN	0.878	0.640	0.999	0.989	0.996	0.932	0.978	0.797	0.915	0.901	0.920	0.852	0.892	0.838
$2$ -cell CCNN	0.854	0.632	0.998	0.975	0.994	0.911	0.973	0.764	0.829	0.893	0.913	0.873	0.769	0.776
CNMTDL	0.878	0.685	0.999	0.991	0.997	0.936	0.985	0.812	0.929	0.901	0.923	0.879	0.932	0.855

Table 3: Comparison of CNMTDL with CCNN variants constructed using only

0

-cell or only

2

-cell frameworks.

Models	Retina		Derma		Blood		OrganC		OrganS		OrganA		Adrenal
Models	AUC	ACC	AUC	ACC	AUC	ACC	AUC	ACC	AUC	ACC	AUC	ACC	AUC	ACC
Curl-CCNN	0.857	0.670	0.953	0.803	0.999	0.991	0.996	0.933	0.981	0.801	0.998	0.951	0.919	0.856
Div-CCNN	0.820	0.608	0.900	0.734	0.999	0.974	0.992	0.897	0.969	0.748	0.992	0.901	0.518	0.769
Curl-Div-CCNN	0.873	0.653	0.955	0.802	0.999	0.990	0.996	0.930	0.982	0.810	0.998	0.956	0.914	0.856
CNMTDL	0.878	0.685	0.955	0.809	0.999	0.991	0.997	0.936	0.986	0.812	0.998	0.959	0.923	0.879

Table 4: Comparison of CNMTDL with models constructed using different Hodge decomposition components.

Figure 4: Performance across dataset scales and classification tasks. (A) AUC and ACC of the top performing models grouped by dataset sizes. (B) AUC and ACC of the top performing models grouped by number of classification task classes.

Figure 5: Illustration of Combinatorial Complex Neural Network. (A) Combinatorial Complex is constructed on the star domain using $0$ -cells (nodes) and $2$ -cells (faces). (B) Neighborhood message passing is performed on each $n$ -cell and message aggregation performed within the adjacent neighbors. (C) The resulting features are propagated through successive CCNN layers, fused and passed to a classifier for the final task.

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TopoCL: Topological Contrastive Learning for Medical Imaging

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Contrastive learning (CL) has become a powerful approach for learning representations from unlabeled images. However, existing CL methods focus predominantly on visual appearance features while neglecting topological characteristics (e.g., connectivity patterns, boundary configurations, cavity formations) that provide valuable cues for medical image analysis. To address this limitation, we propose a new topological CL framework (TopoCL) that explicitly exploits topological structures during contrastive learning for medical imaging. Specifically, we first introduce topology-aware augmentations that control topological perturbations using a relative bottleneck distance between persistence diagrams, preserving medically relevant topological properties while enabling controlled structural variations. We then design a Hierarchical Topology Encoder that captures topological features through self-attention and cross-attention mechanisms. Finally, we develop an adaptive mixture-of-experts (MoE) module to dynamically integrate visual and topological representations. TopoCL can be seamlessly integrated with existing CL methods. We evaluate TopoCL on five representative CL methods (SimCLR, MoCo-v3, BYOL, DINO, and Barlow Twins) and five diverse medical image classification datasets. The experimental results show that TopoCL achieves consistent improvements: an average gain of +3.26% in linear probe classification accuracy with strong statistical significance, verifying its effectiveness.

Abstract

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