On Euclidean Embeddings and Bandwidth Minimization
نویسندگان
چکیده
We study Euclidean embeddings of Euclidean metrics and present the following four results: (1) an O(log n √ log logn) approximation for minimum bandwidth in conjunction with a semi-definite relaxation, (2) an O(log n) approximation in O(n) time using a new constraint set, (3) a lower bound of Θ( √ logn) on the least possible volume distortion for Euclidean metrics, (4) a new embedding with O( √ logn) distortion of point-to-subset distances.
منابع مشابه
Limitations of Learning via Embeddings in Euclidean Half-Spaces
The notion of embedding a class of dichotomies in a class of linear half spaces is central to the support vector machines paradigm. We examine the question of determining the minimal Euclidean dimension and the maximal margin that can be obtained when the embedded class has a finite VC dimension. We show that an overwhelming majority of the family of finite concept classes of any constant VC di...
متن کاملWord Re-Embedding via Manifold Dimensionality Retention
Word embeddings seek to recover a Euclidean metric space by mapping words into vectors, starting from words cooccurrences in a corpus. Word embeddings may underestimate the similarity between nearby words, and overestimate it between distant words in the Euclidean metric space. In this paper, we re-embed pre-trained word embeddings with a stage of manifold learning which retains dimensionality....
متن کاملHybed: Hyperbolic Neural Graph Embedding
Neural embeddings have been used with great success in Natural Language Processing (NLP). They provide compact representations that encapsulate word similarity and attain state-of-the-art performance in a range of linguistic tasks. The success of neural embeddings has prompted significant amounts of research into applications in domains other than language. One such domain is graph-structured d...
متن کاملGreedy Embeddings, Trees, and Euclidean vs. Lobachevsky Geometry
A greedy embedding of an unweighted undirected graph G = (V, E) into a metric space (X, ρ) is a function f : V → X such that for every source-sink pair of different vertices s, t ∈ V it is the case that s has a neighbor v in G with ρ(f(v), f(t)) < ρ(f(s), f(t)). Finding greedy embeddings of connectivity graphs helps to build distributed routing schemes with compact routing tables. In this paper...
متن کاملLeast-Distortion Euclidean Embeddings of Graphs: Products of Cycles and Expanders
Embeddings of finite metric spaces into Euclidean space have been studied in several contexts: The local theory of Banach spaces, the design of approximation algorithms, and graph theory. The emphasis is usually on embeddings with the least possible distortion. That is, one seeks an embedding that minimizes the bi-Lipschitz constant of the mapping. This question has also been asked for embeddin...
متن کامل