Optimal Bounds for Johnson-Lindenstrauss Transformations

TL;DR

This paper establishes the exact asymptotic bounds on the dimension needed for Johnson-Lindenstrauss projections to preserve Euclidean distances, clarifying when such low-dimensional embeddings are possible or impossible.

Contribution

It provides a precise threshold for the dimension of Johnson-Lindenstrauss transformations, delineating the boundary between feasible and infeasible projections.

Findings

Identifies the asymptotic dimension threshold for distance-preserving projections.

Shows the non-existence of such projections below the threshold.

Confirms the existence of projections above the threshold.

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.