# Optimal Bounds for Johnson-Lindenstrauss Transformations

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.

• 5 publications
• 4 publications
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09/03/2020

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We present a simple and efficient acceleration technique for an arbitrar...
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### Convex Optimization Learning of Faithful Euclidean Distance Representations in Nonlinear Dimensionality Reduction

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