Almost-Euclidean subspaces of l1Nvia tensor products: a simple approach to randomness reduction

Authors:
Piotr Indyk;Stanislaw Szarek
Affiliations:
MIT;CWRU & Paris 6
Venue:
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Year:
2010

Citing 6
Cited 2

Asymptotic theory of finite dimensional normed spaces

Asymptotic theory of finite dimensional normed spaces
Dimensionality reduction techniques for proximity problems

SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
Uncertainty principles, extractors, and explicit embeddings of l2 into l1

Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Almost Euclidean subspaces of ℓN1 via expander codes

Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
The geometry of graphs and some of its algorithmic applications

SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
Euclidean Sections of $\ell_1^N$ with Sublinear Randomness and Error-Correction over the Reals

APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques

Deterministic construction of a high dimensional lp section in l1n for any p

Proceedings of the forty-third annual ACM symposium on Theory of computing
From Low-Distortion Norm Embeddings to Explicit Uncertainty Relations and Efficient Information Locking

Journal of the ACM (JACM)

Quantified Score

Hi-index	0.00

Visualization

Abstract

It has been known since 1970's that the N-dimensional l1- space contains almost Euclidean subspaces whose dimension is Ω(N). However, proofs of existence of such subspaces were probabilistic, hence non-constructive, which made the results not-quite-suitable for subsequently discovered applications to high-dimensional nearest neighbor search, error-correcting codes over the reals, compressive sensing and other computational problems. In this paper we present a "low-tech" scheme which, for any γ 0, allows us to exhibit almost Euclidean Ω(N)- dimensional subspaces of l1N while using only Nγ random bits. Our results extend and complement (particularly) recent work by Guruswami-Lee-Wigderson. Characteristic features of our approach include (1) simplicity (we use only tensor products) and (2) yielding almost Euclidean subspaces with arbitrarily small distortions.