Simple, unified analysis of Johnson-Lindenstrauss with applications

We present a simple and unified analysis of the Johnson-Lindenstrauss (JL) lemma, a cornerstone in the field of dimensionality reduction critical for managing high-dimensional data. Our approach not only simplifies the understanding but also unifies various constructions under the JL framework, including spherical, binary-coin, sparse JL, Gaussian and sub-Gaussian models. This simplification and unification make significant strides in preserving the intrinsic geometry of data, essential across diverse applications from streaming algorithms to reinforcement learning. Notably, we deliver the first rigorous proof of the spherical construction's effectiveness and provide a general class of sub-Gaussian constructions within this simplified framework. At the heart of our contribution is an innovative extension of the Hanson-Wright inequality to high dimensions, complete with explicit constants. By employing simple yet powerful probabilistic tools and analytical techniques, such as an enhanced diagonalization process, our analysis not only solidifies the JL lemma's theoretical foundation by removing an independence assumption but also extends its practical reach, showcasing its adaptability and importance in contemporary computational algorithms.