Quasi-isometric embedding of Kerr poloidal submanifolds

A Quasi-isometric Embedding Algorithm

The Whitney embedding theorem gives an upper bound on the smallest embedding dimension of a manifold. If a data set lies on a manifold, a random projection into this reduced dimension will retain the manifold structure. Here we present an algorithm to find a projection that distorts the data as little as possible.

On Isometric and Conformal Rigidity of Submanifolds

Let f, g : Mn → Rn+d be two immersions of an n-dimensional differentiable manifold into Euclidean space. That g is conformal (isometric) to f means that the metrics induced on Mn by f and g are conformal (isometric). We say that f is conformally (isometrically) rigid if given any other conformal (isometric) immersion g there exists a conformal (isometric) diffeomorphism Υ from an open subset of...

Topologically Constrained Isometric Embedding

— We present a new algorithm for nonlinear dimensionality reduction that consistently uses global information, which enables understanding the intrinsic geometry of non-convex manifolds. Compared to methods that consider only local information, our method appears to be more robust to noise. We demonstrate the performance of our algorithm and compare it to state-of-the-art methods on synthetic a...

Isometric Embedding of Riemannian Manifolds

Isometric Embedding and Continuum ISOMAP

Recently, the Isomap algorithm has been proposed for learning a nonlinear manifold from a set of unorganized high-dimensional data points. It is based on extending the classical multidimensional scaling method for dimension reduction. In this paper, we present a continuous version of Isomap which we call continuum isomap and show that manifold learning in the continuous framework is reduced to ...