标题： 预处理以符合不可表示条件 显示英文标题

标题： Preconditioning to comply with the Irrepresentable Condition

摘要： 预处理是一种来自数值线性代数的技术，它可以加速求解方程组的算法。 本文展示了预处理如何规避稀疏线性回归中一致性标志的一个严苛假设。 给定满足标准回归方程的 $X \in R^{n \times p}$ 和 $Y \in R^n$，本文证明了即使设计矩阵 $X$ 不满足Lasso的不可表示条件，设计矩阵 $F X$ 通常也满足该条件，其中 $F \in R^{n\times n}$ 是本文定义的预处理矩阵。 通过在$(F X, F Y)$上计算 Lasso，而不是在$(X, Y)$上计算，对$X$的必要假设变得不那么严格。 我们的预条件器$F$确保设计矩阵的奇异值要么为零，要么为一。 当$n\ge p$时，$F X$的列是正交的，并且预条件器总是避免了严格的假设。 当$p\ge n$时， $F$将设计矩阵投影到Stiefel流形上；$F X$的行是正交的。 我们给出了理论结果和模拟结果，表明在高维情况下，预条件可以帮助规避严格的假设，在线性回归中提高大量模型选择技术的统计性能。 模拟结果尤其令人鼓舞。 显示英文摘要

摘要： Preconditioning is a technique from numerical linear algebra that can accelerate algorithms to solve systems of equations. In this paper, we demonstrate how preconditioning can circumvent a stringent assumption for sign consistency in sparse linear regression. Given $X \in R^{n \times p}$ and $Y \in R^n$ that satisfy the standard regression equation, this paper demonstrates that even if the design matrix $X$ does not satisfy the irrepresentable condition for the Lasso, the design matrix $F X$ often does, where $F \in R^{n\times n}$ is a preconditioning matrix defined in this paper. By computing the Lasso on $(F X, F Y)$, instead of on $(X, Y)$, the necessary assumptions on $X$ become much less stringent. Our preconditioner $F$ ensures that the singular values of the design matrix are either zero or one. When $n\ge p$, the columns of $F X$ are orthogonal and the preconditioner always circumvents the stringent assumptions. When $p\ge n$, $F$ projects the design matrix onto the Stiefel manifold; the rows of $F X$ are orthogonal. We give both theoretical results and simulation results to show that, in the high dimensional case, the preconditioner helps to circumvent the stringent assumptions, improving the statistical performance of a broad class of model selection techniques in linear regression. Simulation results are particularly promising.