A major open problem in the field of metric embedding is the existence of dimension reduction for @math -point subsets of Euclidean space, such that both distortion and dimension depend only on the doubling constant of the pointset, and not on its cardinality. In this paper, we negate this possibility for @math spaces with @math . In particular, we introduce an @math -point subset of @math with doubling constant O(1), and demonstrate that any embedding of the set into @math with distortion @math must have @math .
There are several results on embedding metric spaces with low intrinsic dimension into low dimensional normed space with low distortion: Assouad @cite_12 showed that the snowflakes of doubling metrics For @math , an @math -snowflake of a metric @math is the metric @math , that is, all distances are taken to power @math . embed with constant distortion into constant dimensional Euclidean space. In particular, this is a positive answer to question:d for this special case.
@cite_12
2012238497 2075022748 2053578961 2036959865
We consider the problem of computing the smallest possible distortion for embedding of a given n-point metric space into ℝ d , where d is fixed (and small). For d = 1, it was known that approximating the minimum distortion with a factor better than roughly n 1 12 is NP-hard. From this result we derive inapproximability with a factor roughly n 1 (22d-10) for every fixed d ≥ 2, by a conceptually very simple reduction. However, the proof of correctness involves a nontrivial result in geometric topology (whose current proof is based on ideas due to Jussi Vaisala), For d > 3, we obtain a stronger inapproximability result by a different reduction: assuming P≠NP, no polynomial-time algorithm can distinguish between spaces embeddable in ℝ d with constant distortion from spaces requiring distortion at least n c d , for a constant c > 0. The exponent c d has the correct order of magnitude, since every n-point metric space can be embedded in ℝ d with distortion O(n 2 d log 3 2 n) and such an embedding can be constructed in polynomial time by random projection. For d = 2, we give an example of a metric space that requires a large distortion for embedding in ℝ 2 , while all not too large subspaces of it embed almost isometrically. We devise a new embedding technique, which we call measured descent, based on decomposing a metric space locally, at varying speeds, according to the density of some probability measure. This provides a refined and unified framework for the two primary methods of constructing Frechet embeddings for finite metrics, due to Bourgain (1985) and Rao (1999). We prove that any n-point metric space (X, d) embeds in Hilbert space with distortion (O ( _ X n ) , ) where α X is a geometric estimate on the decomposability of X. As an immediate corollary, we obtain an (O ( ( _ X ) n ) ) distortion embedding, where λ X is the doubling constant of X. Since λ X ≤ n, this result recovers Bourgain’s theorem, but when the metric X is, in a sense, “low-dimensional,” improved bounds are achieved. The doubling dimension of a metric is the smallest k such that any ball of radius 2r can be covered using 2k balls of radius r. This concept for abstract metrics has been proposed as a natural analog to the dimension of a Euclidean space. If we could embed metrics with low doubling dimension into low dimensional Euclidean spaces, they would inherit several algorithmic and structural properties of the Euclidean spaces. Unfortunately however, such a restriction on dimension does not suffice to guarantee embeddibility in a normed space.In this paper we explore the option of bypassing the embedding. In particular we show the following for low dimensional metrics: Quasi-polynomial time (1+e)-approximation algorithm for various optimization problems such as TSP, k-median and facility location. (1+e)-approximate distance labeling scheme with optimal label length. (1+e)-stretch polylogarithmic storage routing scheme. We study the metric properties of finite subsets of L1. The analysis of such metrics is central to a number of important algorithmic problems involving the cut structure of weighted graphs, including the Sparsest Cut Problem, one of the most compelling open problems in the field of approximation algorithms. Additionally, many open questions in geometric non-linear functional analysis involve the properties of finite subsets of L1.We present some new observations concerning the relation of L1 to dimension, topology, and Euclidean distortion. We show that every n-point subset of L1 embeds into L2 with average distortion O(√log n), yielding the first evidence that the conjectured worst-case bound of O(√log n) is valid. We also address the issue of dimension reduction in Lp for p ∈ (1, 2). We resolve a question left open by M. Charikar and A. Sahai [Dimension reduction in the l1 norm, in: Proceedings of the 43rd Annual IEEE Conference on Foundations of Computer Science, ACM, 2002, pp. 251-260] concerning the impossibility of dimension reduction with a linear map in the above cases, and we show that a natural variant of the recent example of Brinkman and Charikar [On the impossibility of dimension reduction in l1, in: Proceedings of the 44th Annual IEEE Conference on Foundations of Computer Science, ACM, 2003, pp. 514-523], cannot be used to prove a lower bound for the non-linear case. This is acomplished by exhibiting constant-distortion embeddings of snowflaked planar metrics into Euclidean space.