Bouligand dimension and almost Lipschitz embeddings

Bouligand Dimension and Almost Lipschitz Embeddings

In this paper we present some new properties of the metric dimension defined by Bouligand in 1928 and prove the following new projection theorem: Let dimb(A − A) denote the Bouligand dimension of the set A − A of differences between elements of A. Given any compact set A ⊆ R such that dimb(A−A) < m, then almost every orthogonal projection P of A of rank m is injective on A and P |A has Lipschit...

Almost Bi-lipschitz Embeddings and Almost Homogeneous Sets

This paper is concerned with embeddings of homogeneous spaces into Euclidean spaces. We show that any homogeneous metric space can be embedded into a Hilbert space using an almost bi-Lipschitz mapping (biLipschitz to within logarithmic corrections). The image of this set is no longer homogeneous, but ‘almost homogeneous’. We therefore study the problem of embedding an almost homogeneous subset ...

Metric Embeddings and Lipschitz Extensions

Quantitative Bi-Lipschitz embeddings of manifolds

The goal of this write up is to provide a more detailed proof of the collapsing result already contained in [2]. While reading this paper the author struggled to understand many details of the proofs, particularly where the details of an argument were spread over several papers or simply assumed obvious. Nothing new is proven in this write up, but results are explicated and an attempt is made t...

Energy-aware adaptive bi-Lipschitz embeddings

We propose a dimensionality reducing matrix design based on training data with constraints on its Frobenius norm and number of rows. Our design criteria is aimed at preserving the distances between the data points in the dimensionality reduced space as much as possible relative to their distances in original data space. This approach can be considered as a deterministic Bi-Lipschitz embedding o...