Topology-Aware Contrastive Learning: Regulating Representation Connectivity via Persistent Homology

Standard contrastive learning minimizes geometric distance between positive pairs, implicitly assuming that strict compactness optimizes discrimination. However, this topology-agnostic confusion neglects intrinsic data structures and topological complexity, leading to class confusion—particularly when aggressive augmentations induce semantic drift. To address this, we propose Topology-Aware Contrastive Learning, a framework that shifts the objective from geometric singularity to topological connectivity. Leveraging Persistent Homology, we explicitly regulate the connectivity of the latent space, ensuring positive pairs maintain an $\alpha\text{--}\beta$ connectivity that balances intra-class cohesion with separability. Theoretically, we formally define the topology-agnostic confusion phenomenon, prove that excessive compactness strictly lower-bounds the probability of confusion and derive a generalization bound demonstrating that richer topological connectivity tightens downstream risk. Furthermore, we establish a measure-theoretic framework to mitigating the sensitivity of our method against varying augmentation strengths. Empirical results on benchmarks confirm that our approach enhances representation quality and reduces reliance on specific augmentation strategies compared to standard baselines.

How do machine learning models learn to recognize images without human labels? They typically try to group similar pictures as tightly together as possible. However, this ignores the natural underlying "shape" of the data. When images are heavily modified during training, this over-tightening causes "class confusion," where the model starts mixing up visually similar but distinct objects—like mistakenly blurring the lines between a bicycle and a motorcycle. To solve this, we introduced Topology-Aware Contrastive Learning (TACL). Instead of crushing similar images into a single microscopic point, our framework acts like a mapmaker. It learns the broader structural connections within the data. This allows the system to keep similar items logically connected while maintaining clear, healthy boundaries between different categories. We mathematically proved that forcing too much compactness directly causes this confusion. By preserving the natural structural connections instead, our approach helps algorithms avoid these mix-ups. This makes the learned representations much more robust and reliable when handling complex, real-world visual data, without relying on specific training tricks.