Distance Preserving Embeddings for General n-Dimensional Manifolds
نویسنده
چکیده
Low dimensional embeddings of manifold data have gained popularity in the last decade. However, a systematic finite sample analysis of manifold embedding algorithms largely eludes researchers. Here we present two algorithms that embed a general n-dimensional manifold intoR (where d only depends on some key manifold properties such as its intrinsic dimension, volume and curvature) that guarantee to approximately preserve all interpoint geodesic distances.
منابع مشابه
VOLUME PRESERVING EMBEDDINGS OF OPEN SUBSETS OF R n INTO MANIFOLDS
We consider a connected smooth n-dimensional manifold M endowed with a volume form Ω, and we show that an open subset U of Rn of Lebesgue measure Vol(U) embeds into M by a smooth volume preserving embedding whenever the volume condition Vol(U) ≤ Vol(M,Ω) is met.
متن کاملA Geometry Preserving Kernel over Riemannian Manifolds
Abstract- Kernel trick and projection to tangent spaces are two choices for linearizing the data points lying on Riemannian manifolds. These approaches are used to provide the prerequisites for applying standard machine learning methods on Riemannian manifolds. Classical kernels implicitly project data to high dimensional feature space without considering the intrinsic geometry of data points. ...
متن کاملEMBEDDINGS OF k - CONNECTED n - MANIFOLDS INTO
We obtain estimations for isotopy classes of embeddings of k-connected nmanifolds into R2n−k−1 for n ≥ 2k+6 and k ≥ 0. This is done in terms of an exact sequence involving the Whitney invariants and an explicitly constructed action of Hk+1(N ;Z2) on the set of embeddings. The proof involves reduction to classification of embeddings of punctured manifold and uses parametric connected sum of embe...
متن کاملConformal mappings preserving the Einstein tensor of Weyl manifolds
In this paper, we obtain a necessary and sufficient condition for a conformal mapping between two Weyl manifolds to preserve Einstein tensor. Then we prove that some basic curvature tensors of $W_n$ are preserved by such a conformal mapping if and only if the covector field of the mapping is locally a gradient. Also, we obtained the relation between the scalar curvatures of the Weyl manifolds r...
متن کاملParametric Manifold Learning Via Sparse Multidimensional Scaling
We propose a metric-learning framework for computing distance-preserving maps that generate low-dimensional embeddings for a certain class of manifolds. We employ Siamese networks to solve the problem of least squares multidimensional scaling for generating mappings that preserve geodesic distances on the manifold. In contrast to previous parametric manifold learning methods we show a substanti...
متن کامل