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.

翻译：我们提出了一种简化且统一的Johnson-Lindenstrauss（JL）引理分析，该引理是处理高维数据时降维技术的基石。我们的方法简化了理解过程，并在JL框架下统一了多种构造方案，包括球形构造、二进制硬币构造、稀疏JL构造、高斯模型以及次高斯模型。这种统一保持了数据的内在几何结构，这对于从流算法到强化学习的各类应用至关重要。我们首次给出了球形构造有效性的严格证明，并在此简化框架内引入了一类通用的次高斯构造。我们贡献的核心在于将Hanson-Wright不等式创新性地扩展到高维情形，并提供了明确的常数。通过运用简单而强大的概率工具与分析技术（例如改进的对角化过程），我们的分析消除了独立性假设，从而巩固了JL引理的理论基础，并将其实际应用范围扩展至当代算法。