标题： 高维情况下的变量选择一致性紧条件 显示英文标题

标题： Tight conditions for consistency of variable selection in the context of high dimensionality

摘要： 我们解决在高环境维数的回归模型中的变量选择问题，即当变量数量非常大时的情况。 主要关注的是相关变量的数量，称为内在维数，远小于环境维数d的情况。 在不假设潜在回归函数的任何参数形式的情况下，我们得到了紧密条件，使得可以一致地估计相关变量的集合。 这些条件将内在维数与环境维数以及样本量联系起来。 在这些紧密条件下可证明是一致的程序是基于将经验傅里叶系数的二次泛函与适当选择的阈值进行比较。 渐近分析揭示了存在两种相当不同的情形。 第一种情形是当内在维数固定时。 在这种情况下，非参数回归的情况与线性回归相同，即只有当log d相比样本量n较小时，才能进行一致的变量选择。 在第二种情形中情况不同，即当由s表示的相关变量数量随着$n\to\infty$趋于无穷大时。 然后我们证明，在非参数设置中，只有当s+loglog d相比log n较小时，才能进行一致的变量选择。 我们将这些结果应用于推导变量选择问题的最小最大分离率。 显示英文摘要

摘要： We address the issue of variable selection in the regression model with very high ambient dimension, that is, when the number of variables is very large. The main focus is on the situation where the number of relevant variables, called intrinsic dimension, is much smaller than the ambient dimension d. Without assuming any parametric form of the underlying regression function, we get tight conditions making it possible to consistently estimate the set of relevant variables. These conditions relate the intrinsic dimension to the ambient dimension and to the sample size. The procedure that is provably consistent under these tight conditions is based on comparing quadratic functionals of the empirical Fourier coefficients with appropriately chosen threshold values. The asymptotic analysis reveals the presence of two quite different re gimes. The first regime is when the intrinsic dimension is fixed. In this case the situation in nonparametric regression is the same as in linear regression, that is, consistent variable selection is possible if and only if log d is small compared to the sample size n. The picture is different in the second regime, that is, when the number of relevant variables denoted by s tends to infinity as $n\to\infty$. Then we prove that consistent variable selection in nonparametric set-up is possible only if s+loglog d is small compared to log n. We apply these results to derive minimax separation rates for the problem of variable