Goal-Oriented Euclidean Heuristics with Manifold Learning

On images of continuous functions from a compact manifold to Euclidean space

Shamap: Shape-based Manifold Learning

For manifold learning, it is assumed that high-dimensional sample/data points are on an embedded low-dimensional manifold. Usually, distances among samples are computed to represent the underlying data structure, for a specified distance measure such as the Euclidean distance or geodesic distance. For manifold learning, here we propose a metric according to the angular change along a geodesic l...

Data-based Manifold Reconstruction via Tangent Bundle Manifold Learning

The goal of Manifold Learning (ML) is to find a description of low-dimensional structure of an unknown q-dimensional manifold embedded in high-dimensional ambient Euclidean space R p , q < p, from their finite samples. There are a variety of formulations of the problem. The methods of Manifold Approximation (MA) reconstruct (estimate) the manifold but don’t find a low-dimensional parameterizati...

Curvature-aware Manifold Learning

Traditional manifold learning algorithms assumed that the embedded manifold is globally or locally isometric to Euclidean space. Under this assumption, they divided manifold into a set of overlapping local patches which are locally isometric to linear subsets of Euclidean space. By analyzing the global or local isometry assumptions it can be shown that the learnt manifold is a flat manifold wit...

Tangent Bundle of the Hypersurfaces in a Euclidean Space

Let $M$ be an orientable hypersurface in the Euclidean space $R^{2n}$ with induced metric $g$ and $TM$ be its tangent bundle. It is known that the tangent bundle $TM$ has induced metric $overline{g}$ as submanifold of the Euclidean space $R^{4n}$ which is not a natural metric in the sense that the submersion $pi :(TM,overline{g})rightarrow (M,g)$ is not the Riemannian submersion. In this paper...