@inproceedings{72c8d9f78c8a4ae9aa53d7278d55769d,
title = "Optimal compression of approximate inner products and dimension reduction",
abstract = "Let X be a set of n points of norm at most 1 in the Euclidean space R^k, and suppose ϵ 0. An ϵ -distance sketch for X is a data structure that, given any two points of X enables one to recover the square of the (Euclidean) distance between them up to an additive} error of ϵ . Let f(n,k,ϵ ) denote the minimum possible number of bits of such a sketch. Here we determine f(n,k,ϵ ) up to a constant factor for all n ϵ k ϵ 1 and all ϵ ϵ \frac{1}{n^{0.49}}. Our proof is algorithmic, and provides an efficient algorithm for computing a sketch of size O(f(n,k,ϵ )/n) for each point, so that the square of the distance between any two points can be computed from their sketches up to an additive error of ϵ in time linear in the length of the sketches. We also discuss the case of smaller ϵ 2/√ n and obtain some new results about dimension reduction in this range.",
keywords = "Gaussian correlation, compression scheme, dimension reduction, epsilon-net",
author = "Noga Alon and Bo'az Klartag",
note = "Publisher Copyright: {\textcopyright} 2017 IEEE.; 58th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2017 ; Conference date: 15-10-2017 Through 17-10-2017",
year = "2017",
month = nov,
day = "10",
doi = "10.1109/FOCS.2017.65",
language = "الإنجليزيّة",
isbn = "9781538634646",
series = "Annual Symposium on Foundations of Computer Science - Proceedings",
publisher = "IEEE Computer Society",
pages = "639--650",
booktitle = "Proceedings - 58th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2017",
address = "الولايات المتّحدة",
}