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 .
The first impossibility result on dimension reduction in @math is due to @cite_48 , who showed that there exists an @math -point subset of @math that requires @math for any @math -distortion embedding to @math . Following their work, there have been many different proofs and extensions of this result, using various techniques. The original @cite_48 argument was based on linear programming and duality, then @cite_44 gave a geometric proof. For the @math distortion regime, @cite_20 used combinatorial techniques to show that the dimension must be at least @math (and also gave a different proof of the original result). Recently @cite_39 applied an information theoretic argument to reprove the results of @cite_48 @cite_20 . As for linear dimension reductions, @cite_38 showed a strong lower bound for @math with @math .
@cite_38 @cite_48 @cite_39 @cite_44 @cite_20
2036959865 2001085501 2150285483 2149463264
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. Given a set of @math points in @math , how many dimensions are needed to represent all pair wise distances within a specific distortion? This dimension-distortion tradeoff question is well understood for the @math norm, where @math dimensions suffice to achieve @math distortion. In sharp contrast, there is a significant gap between upper and lower bounds for dimension reduction in @math . A recent result shows that distortion @math can be achieved with @math dimensions. On the other hand, the only lower bounds known are that distortion @math requires @math dimensions and that distortion @math requires @math dimensions. In this work, we show the first near linear lower bounds for dimension reduction in @math . In particular, we show that @math distortion requires at least @math dimensions. Our proofs are combinatorial, but inspired by linear programming. In fact, our techniques lead to a simple combinatorial argument that is equivalent to the LP based proof of Brinkman-Charikar for lower bounds on dimension reduction in @math . We develop a framework for proving approximation limits of polynomial size linear programs (LPs) from lower bounds on the nonnegative ranks of suitably defined matrices. This framework yields unconditional impossibility results that are applicable to any LP as opposed to only programs generated by hierarchies. Using our framework, we prove that O(n1 2-ϵ)-approximations for CLIQUE require LPs of size 2nΩ(ϵ). This lower bound applies to LPs using a certain encoding of CLIQUE as a linear optimization problem. Moreover, we establish a similar result for approximations of semidefinite programs by LPs. Our main technical ingredient is a quantitative improvement of Razborov’s [38] rectangle corruption lemma for the high error regime, which gives strong lower bounds on the nonnegative rank of shifts of the unique disjointness matrix. The Johnson--Lindenstrauss lemma shows that any n points in Euclidean space (i.e., ℝn with distances measured under the e 2 norm) may be mapped down to O((log n) e2) dimensions such that no pairwise distance is distorted by more than a (1 p e) factor. Determining whether such dimension reduction is possible in e 1 has been an intriguing open question. We show strong lower bounds for general dimension reduction in e 1 . We give an explicit family of n points in e 1 such that any embedding with constant distortion D requires nΩ(1 D2) dimensions. This proves that there is no analog of the Johnson--Lindenstrauss lemma for e 1 ; in fact, embedding with any constant distortion requires nΩ(1) dimensions. Further, embedding the points into e 1 with (1pe) distortion requires nh−O(e log(1 e)) dimensions. Our proof establishes this lower bound for shortest path metrics of series-parallel graphs. We make extensive use of linear programming and duality in devising our bounds. We expect that the tools and techniques we develop will be useful for future investigations of embeddings into e 1 .