A Simple proof of Johnson-Lindenstrauss extension theorem

TL;DR

This paper simplifies the proof of the Johnson-Lindenstrauss extension theorem, removing the need for dimension reduction and the Kirszbraun theorem, thus making the extension process more straightforward.

Contribution

The authors provide a simplified proof of the Johnson-Lindenstrauss extension theorem that avoids complex dimension reduction techniques and the Kirszbraun theorem.

Findings

Simplified proof of the extension theorem

Avoids dimension reduction techniques

Eliminates reliance on Kirszbraun theorem

Abstract

Johnson and Lindenstrauss proved that any Lipschitz mapping from an $n$-point subset of a metric space into Hilbert space can be extended to the whole space, while increasing the Lipschitz constant by a factor of $O(\sqrt{\log n})$. We present a simplification of their argument that avoids dimension reduction and the Kirszbraun theorem.