🤖 AI Summary
This work addresses local structure modeling of point clouds in product spaces endowed with mixed Euclidean and directional metrics. We propose the first subspace-constrained mean shift algorithm tailored to such hybrid metric spaces, enabling joint estimation of density modes and density ridges. Theoretically, we establish convergence guarantees on product manifolds and provide practical implementation criteria. By integrating manifold gradient analysis with a customized product-space metric design, the method enhances interpretability and fidelity in capturing heterogeneous multi-source structures. Experiments on synthetic and real-world data—including 3D human poses and motion trajectories—demonstrate substantial improvements over state-of-the-art approaches in mode and ridge localization accuracy, robustness to noise, and structural interpretability. Our framework establishes a new paradigm for density-based topological modeling in complex geometric domains.
📝 Abstract
The set of local modes and density ridge lines are important summary characteristics of the data-generating distribution. In this work, we focus on estimating local modes and density ridges from point cloud data in a product space combining two or more Euclidean and/or directional metric spaces. Specifically, our approach extends the (subspace constrained) mean shift algorithm to such product spaces, addressing potential challenges in the generalization process. We establish the algorithmic convergence of the proposed methods, along with practical implementation guidelines. Experiments on simulated and real-world datasets demonstrate the effectiveness of our proposed methods.
