This paper revisits the performance of Rademacher random projections, establishing novel statistical guarantees that are numerically sharp and non-oblivious with respect to the input data. More specifically, the central result is the Schur-concavity property of Rademacher random projections with respect to the inputs. This offers a novel geometric perspective on the performance of random projections, while improving quantitatively on bounds from previous works. As a corollary of this broader result, we obtained the improved performance on data which is sparse or is distributed with small spread. This non-oblivious analysis is a novelty compared to techniques from previous work, and bridges the frequently observed gap between theory and practise. The main result uses an algebraic framework for proving Schur-concavity properties, which is a contribution of independent interest and an elegant alternative to derivative-based criteria.

翻译：本文重新审视了Rademacher随机投影的性能，建立了在数值上精确且对输入数据非盲的新统计保证。具体而言，核心成果是Rademacher随机投影相对于输入具有Schur-凹性。这一性质不仅为随机投影的性能提供了新颖的几何视角，还在定量上改进了以往工作的界值。作为这一更广泛结果的推论，我们获得了在稀疏分布或小散布数据上的改进性能。与以往工作方法相比，这种非盲分析具有创新性，并弥合了理论与实践中常见的差距。主要结果采用代数框架证明Schur-凹性，这不仅是一项具有独立意义的贡献，也是基于导数准则的优雅替代方案。