Simple, unified analysis of Johnson-Lindenstrauss with applications
Yingru Li
TL;DR
This work develops a simple, unified probabilistic framework for Johnson-Lindenstrauss (JL) dimensionality reduction that encompasses spherical, Gaussian, binary-coin, sparse JL, and general sub-Gaussian constructions. A key technical advance is a high-dimensional Hanson-Wright inequality with explicit constants, enabling JL analysis without independence assumptions and including a first rigorous proof for the spherical construction. The authors also introduce a general sub-Gaussian class without unit-norm, extend the hdHW tool to non-negative diagonal and to a generalized form with entrywise scaling, and demonstrate applications to uncertainty estimation and covariance factorization in streaming and reinforcement learning settings. The resulting bounds show $m = O(\varepsilon^{-2} \log(1/\delta))$ for the reduced dimension and provide concrete tools for modern algorithms with finite-sample guarantees, explaining practical performance and broadening JL's applicability. Overall, the paper strengthens the theoretical foundations of JL while expanding its utility to contemporary data-driven tasks with rigorous, explicit constants.
Abstract
We present a simplified and unified analysis of the Johnson-Lindenstrauss (JL) lemma, a cornerstone of dimensionality reduction for managing high-dimensional data. Our approach simplifies understanding and unifies various constructions under the JL framework, including spherical, binary-coin, sparse JL, Gaussian, and sub-Gaussian models. This unification preserves the intrinsic geometry of data, essential for applications from streaming algorithms to reinforcement learning. We provide the first rigorous proof of the spherical construction's effectiveness and introduce a general class of sub-Gaussian constructions within this simplified framework. Central to our contribution is an innovative extension of the Hanson-Wright inequality to high dimensions, complete with explicit constants. By using simple yet powerful probabilistic tools and analytical techniques, such as an enhanced diagonalization process, our analysis solidifies the theoretical foundation of the JL lemma by removing an independence assumption and extends its practical applicability to contemporary algorithms.