The LASSO is a recent technique for variable selection in the regression model \bean y & = & X\beta + z, \eean where $X\in \R^{n\times p}$ and $z$ is a centered gaussian i.i.d. noise vector $\mathcal N(0,\sigma^2I)$. The LASSO has been proved to achieve remarkable properties such as exact support recovery of sparse vectors when the columns are sufficently incoherent and low prediction error under even less stringent conditions. However, many matrices do not satisfy small coherence in practical applications and the LASSO estimator may thus suffer from what is known as the slow rate regime. The goal of the present paper is to study the LASSO from a slightly different perspective by proposing a mixture model for the design matrix which is able to capture in a natural way the potentially clustered nature of the columns in many practical situations. In this model, the columns of the design matrix are drawn from a Gaussian mixture model. Instead of requiring incoherence for the design matrix $X$, we only require incoherence of the much smaller matrix of the mixture's centers. Our main result states that $X\beta$ can be estimated with the same precision as for incoherent designs except for a correction term depending on the maximal variance in the mixture model.

翻译：LASSO是一种用于回归模型变量选择的近期技术，其模型为 \bean y & = & X\beta + z, \eean 其中 $X\in \R^{n\times p}$，$z$ 是中心化的高斯独立同分布噪声向量 $\mathcal N(0,\sigma^2I)$。LASSO已被证明在列足够不相关时能实现稀疏向量的精确支撑恢复，并在更宽松条件下具有低预测误差等显著特性。然而，实际应用中许多矩阵不满足低相干性，因此LASSO估计量可能陷入所谓的慢速率区域。本文旨在从略微不同的视角研究LASSO，提出一种针对设计矩阵的混合模型，该模型能自然捕捉许多实际情境中列潜在的聚类特性。在此模型中，设计矩阵的列从高斯混合模型中抽取。我们无需要求设计矩阵 $X$ 具有不相关性，仅需要求远小于它的混合中心矩阵具有不相关性。主要结果表明，除取决于混合模型中最大方差的修正项外，$X\beta$ 的估计精度可与不相关设计的情形相当。