The Johnson-Lindenstrauss (JL) theorem states that a set of points in high-dimensional space can be embedded into a lower-dimensional space while approximately preserving pairwise distances with high probability Johnson and Lindenstrauss (1984). The standard JL theorem uses dense random matrices with Gaussian entries. However, for some applications, sparse random matrices are preferred as they allow for faster matrix-vector multiplication. I outline the constructions and proofs introduced by Achlioptas (2003) and the contemporary standard by Kane and Nelson (2014). Further, I implement and empirically compare these sparse constructions with standard Gaussian JL matrices.

翻译：Johnson-Lindenstrauss (JL) 定理指出，高维空间中的点集可以嵌入到低维空间中，同时以高概率近似保持点对距离 Johnson and Lindenstrauss (1984)。标准的 JL 定理使用具有高斯项的稠密随机矩阵。然而，在某些应用中，稀疏随机矩阵更受青睐，因为它们允许更快的矩阵-向量乘法。本文概述了 Achlioptas (2003) 提出的构造与证明方法，以及 Kane and Nelson (2014) 提出的当代标准方法。此外，本文实现了这些稀疏构造方法，并通过实验将其与标准高斯 JL 矩阵进行了比较。