Persistent homology via curvature-adaptive wing complexes

Persistent homology is a key tool in topological data analysis, used to capture the topological features of data across multiple scales. To accurately capture the topology of data with varying geometric structures, one should construct suitable simplicial complexes. In this paper, we propose wing complexes as a novel method for finite sets of points sampled from a smooth plane curve. The key innovation of the wing complex lies in its ability to stretch and shrink along the tangent and normal directions based on a specific parameter, allowing it to adapt to the local curvature variations of the data. We theoretically derive the topological properties of wing complexes and conduct related experiments. The results demonstrate that, compared to traditional methods, wing complexes provide more persistent and accurate topological features, particularly in regions with rapid local curvature variations, whereas traditional methods often fail to capture the correct topological features.