The Sharp Dimension Bound in the Johnson--Lindenstrauss Lemma

2026-08-13

AI Summary

This study resolves the longstanding conjecture regarding the optimal target dimension in the Johnson-Lindenstrauss lemma across the full parameter range. By synthesizing high-dimensional geometric embedding theory with refined analysis of linear map constructions, this work confirms the Larsen-Nelson conjecture. The primary contribution lies in definitively establishing the optimal target dimension for the JL lemma and rigorously proving that linear maps achieve the theoretical lower bound, which precisely matches existing nonlinear lower bounds. Consequently, this research establishes the optimality of linear mappings for high-dimensional data dimensionality reduction, thereby providing a complete theoretical foundation for the field. These findings settle a fundamental open problem and validate the efficacy of linear projections in preserving metric structure within compressed spaces.

Abstract

The Johnson--Lindenstrauss lemma asserts that every set of $n$ points in $d$-dimensional Euclidean space embeds into $O(\varepsilon^{-2}\log n)$-dimensional Euclidean space with distortion at most $1+\varepsilon$. Larsen and Nelson conjectured that the optimal target dimension throughout the full range of the parameters $n,d, \varepsilon$ is \[ Θ\left(\min\left\{d,n-1,\frac{\log(2+\varepsilon^2n)}{\varepsilon^2}\right\}\right). \] We resolve this conjecture in the affirmative. In fact, we prove the stronger statement that the upper bound is attained by a linear map. The matching lower bound, due to Larsen--Nelson and Alon--Klartag, holds even for nonlinear embeddings.

Problem

• Johnson-Lindenstrauss lemma
• dimension bound
• optimal target dimension
• low-dimensional embedding

Innovation

• Johnson-Lindenstrauss lemma
• Sharp dimension bound
• Linear embedding
• Optimal target dimension
• Dimensionality reduction