Research on Manifold approximation part9(Advanced Machine Learning)
- Uniform Manifold Approximation with Two-phase Optimization
Authors: Hyeon Jeon, Hyung-Kwon Ko, Soohyun Lee, Jaemin Jo, Jinwook Seo
Abstract: We introduce Uniform Manifold Approximation with Two-phase Optimization (UMATO), a dimensionality reduction (DR) technique that improves UMAP to capture the global structure of high-dimensional data more accurately. In UMATO, optimization is divided into two phases so that the resulting embeddings can depict the global structure reliably while preserving the local structure with sufficient accuracy. In the first phase, hub points are identified and projected to construct a skeletal layout for the global structure. In the second phase, the remaining points are added to the embedding preserving the regional characteristics of local areas. Through quantitative experiments, we found that UMATO (1) outperformed widely used DR techniques in preserving the global structure while (2) producing competitive accuracy in representing the local structure. We also verified that UMATO is preferable in terms of robustness over diverse initialization methods, number of epochs, and subsampling techniques.
2. Approximating Riemannian manifolds by polyhedra
Authors: Daniel Meyer, Eric Toubiana
Abstract: This is a study on approximating a Riemannian manifold by polyhedra. Our scope is understanding Tullio Regge’s [52] article in the restricted Riemannian frame. We give a proof of the Regge theorem along lines close to its original intuition: one can approximate a compact domain of a Riemannian manifold by polyhedra in such a way that the integral of the scalar curvature is approximated by a corresponding polyhedral curvature
