On Sparsity and Sub-Gaussianity in the Johnson-Lindenstrauss Lemma

Aurélien GarivierUMPA-ENSL, MC2, Emmanuel PilliatUMPA-ENSL
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Abstract

We provide a simple proof of the Johnson-Lindenstrauss lemma for sub-Gaussian variables. We extend the analysis to identify how sparse projections can be, and what the cost of sparsity is on the target dimension.The Johnson-Lindenstrauss lemma is the theoretical core of the dimensionality reduction methods based on random projections. While its original formulation involves matrices with Gaussian entries, the computational cost of random projections can be drastically reduced by the use of simpler variables, especially if they vanish with a high probability. In this paper, we propose a simple and elementary analysis of random projections under classical assumptions that emphasizes the key role of sub-Gaussianity. Furthermore, we show how to extend it to sparse projections, emphasizing the limits induced by the sparsity of the data itself.
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论约翰逊-林登斯特劳斯定理中的稀疏性和次高斯性
我们为亚高斯变量的约翰逊-林登斯特劳斯(Johnson-Lindenstrauss)lemma 提供了一个简单的证明。约翰逊-林登斯特劳斯定理是基于随机投影的降维方法的理论核心。约翰逊-林登斯特劳斯训令是基于随机投影的降维方法的理论核心。虽然它的原始公式涉及具有高斯条目的矩阵,但通过使用更简单的变量,尤其是当它们以高概率消失时,可以大大降低随机投影的计算成本。本文提出了经典假设下随机投影的简单基本分析,强调了亚高斯性的关键作用。此外,我们还展示了如何将其扩展到稀疏投影,强调数据本身的稀疏性所引起的限制。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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