Combinatorial Network-Based Manifold Topological Deep Learning for Image Analysis

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.

Topological data analysis (TDA), particularly persistent homology (PH), captures geometric structural properties in medical images (e.g., connected components, loops, shape characteristics), which conventional pixel-level deep learning approaches often neglect. While many topological descriptors are known for converting persistence diagrams (PDs) or raw images into topological feature vectors, existing methods mostly default to a single fixed descriptor (e.g., persistence images), leaving the diversity of topological representations largely unexplored. To the best of our knowledge, there is no known large language model (LLM)-based agentic framework that can automatically determine the most suitable topological descriptors for a given image dataset and produce the corresponding topological feature vectors for downstream tasks. To fill this gap, we propose TopoAgent, an LLM-based agentic framework that automates topology learning for medical image analysis.TopoAgent operates through a Perception--Reasoning--Action--Reflection loop supported by 21 domain-specific tools and dual memory that accumulates experience across runs. Its skill set is distilled from systematic evaluation of 15 topological descriptors across 26 datasets with six classifiers. TopoAgent analyzes input images and their topological characteristics, reasons about which topological descriptors best suit the input, and determines the optimal descriptor and its configuration, all without task-specific training.

Medical image anomaly detection faces unique challenges due to subtle, heterogeneous anomalies embedded in complex anatomical structures. Through systematic Grad-CAM analysis, we reveal that discriminative activation maps fail on medical data, unlike their success on industrial datasets, motivating the need for manifold-level modeling. We propose PDD (Manifold-Prior Diverse Distillation), a framework that unifies dual-teacher priors into a shared high-dimensional manifold and distills this knowledge into dual students with complementary behaviors. Specifically, frozen VMamba-Tiny and wide-ResNet50 encoders provide global contextual and local structural priors, respectively. Their features are unified through a Manifold Matching and Unification (MMU) module, while an Inter-Level Feature Adaption (InA) module enriches intermediate representations. The unified manifold is distilled into two students: one performs layer-wise distillation via InA for local consistency, while the other receives skip-projected representations through a Manifold Prior Affine (MPA) module to capture cross-layer dependencies. A diversity loss prevents representation collapse while maintaining detection sensitivity. Extensive experiments on multiple medical datasets demonstrate that PDD significantly outperforms existing state-of-the-art methods, achieving improvements of up to 11.8%, 5.1%, and 8.5% in AUROC on HeadCT, BrainMRI, and ZhangLab datasets, respectively, and 3.4% in F1 max on the Uni-Medical dataset, establishing new state-of-the-art performance in medical image anomaly detection. The implementation will be released at https://github.com/OxygenLu/PDD