论文标题
高维回归的随机测试：一种更高效，更强大的解决方案
Randomized tests for high-dimensional regression: A more efficient and powerful solution
论文作者
论文摘要
当特征尺寸$ p $与观测值$ n $相称时，我们研究了在高维回归模型中测试全球空的问题的问题。尽管有许多先前研究此问题的工作，但是否存在一种模型不合时宜的测试，计算和享有高功率的高效，仍然保持不安。在本文中，我们通过利用随机投影技术来回答这个问题，并提出了将经典$ f $检验与随机投影步骤融合的测试程序。当与投影维度的系统选择结合使用时，事实证明，所提出的过程是最小的最佳选择，同时，减少了计算和数据存储要求。当基础特征矩阵表现出内在的较低尺寸结构（例如近似块结构或具有指数/多项式特征 - 达到）时，我们在各种情况下说明了我们的结果，事实证明，所提出的测试可以实现尖锐的自适应速率。我们的理论发现通过与合成数据的其他最新测试进行比较，进一步验证。
We investigate the problem of testing the global null in the high-dimensional regression models when the feature dimension $p$ grows proportionally to the number of observations $n$. Despite a number of prior work studying this problem, whether there exists a test that is model-agnostic, efficient to compute and enjoys high power, still remains unsettled. In this paper, we answer this question in the affirmative by leveraging the random projection techniques, and propose a testing procedure that blends the classical $F$-test with a random projection step. When combined with a systematic choice of the projection dimension, the proposed procedure is proved to be minimax optimal and, meanwhile, reduces the computation and data storage requirements. We illustrate our results in various scenarios when the underlying feature matrix exhibits an intrinsic lower dimensional structure (such as approximate block structure or has exponential/polynomial eigen-decay), and it turns out that the proposed test achieves sharp adaptive rates. Our theoretical findings are further validated by comparisons to other state-of-the-art tests on the synthetic data.
