Optimal Bounds for Johnson-Lindenstrauss Transformations
Discrete Mathematics
2018-03-15 v1 Probability
Abstract
In 1984, Johnson and Lindenstrauss proved that any finite set of data in a high-dimensional space can be projected to a lower-dimensional space while preserving the pairwise Euclidean distance between points up to a bounded relative error. If the desired dimension of the image is too small, however, Kane, Meka, and Nelson (2011) and Jayram and Woodruff (2013) independently proved that such a projection does not exist. In this paper, we provide a precise asymptotic threshold for the dimension of the image, above which, there exists a projection preserving the Euclidean distance, but, below which, there does not exist such a projection.
Keywords
Cite
@article{arxiv.1803.05350,
title = {Optimal Bounds for Johnson-Lindenstrauss Transformations},
author = {Michael Burr and Shuhong Gao and Fiona Knoll},
journal= {arXiv preprint arXiv:1803.05350},
year = {2018}
}