Simple, unified analysis of Johnson-Lindenstrauss with applications
This work addresses foundational theoretical gaps in dimensionality reduction for researchers and practitioners in machine learning and data science, though it is incremental in refining existing frameworks.
The paper tackles the problem of simplifying and unifying the analysis of the Johnson-Lindenstrauss lemma for dimensionality reduction, providing a rigorous proof for spherical constructions and introducing a general class of sub-Gaussian models with explicit constants.
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.