标题： 高维逼近回归模型的Lasso一致选择 显示英文标题

标题： Consistent selection via the Lasso for high dimensional approximating regression models

摘要： 在本文中，我们通过流行的Lasso方法研究回归模型中选择的一致性。 在这里，我们偏离传统的线性回归假设，考虑回归函数$f$与给定字典中的$M$个函数的近似。 一致性的目标是该字典中那些函数的索引集，这些函数在所有属于以$f$为中心、半径为$r_{n,M}^2$的$L_2$球内的线性组合中实现对$f$最简洁的近似。 在这个框架中，我们证明可以通过数据相关的惩罚和调参序列$r_{n,M}>\sqrt{\log(Mn)/n}$的$\ell_1$惩罚最小二乘法得到该指标集的一致估计，其中$n$是样本量。 我们的结果适用于任何$1\leq M\leq n^{\gamma}$，对于任何$\gamma>0$。 显示英文摘要

摘要： In this article we investigate consistency of selection in regression models via the popular Lasso method. Here we depart from the traditional linear regression assumption and consider approximations of the regression function $f$ with elements of a given dictionary of $M$ functions. The target for consistency is the index set of those functions from this dictionary that realize the most parsimonious approximation to $f$ among all linear combinations belonging to an $L_2$ ball centered at $f$ and of radius $r_{n,M}^2$. In this framework we show that a consistent estimate of this index set can be derived via $\ell_1$ penalized least squares, with a data dependent penalty and with tuning sequence $r_{n,M}>\sqrt{\log(Mn)/n}$, where $n$ is the sample size. Our results hold for any $1\leq M\leq n^{\gamma}$, for any $\gamma>0$.