关于$\ell_1$ -惩罚稀疏精度矩阵估计的不一致性
On the inconsistency of $\ell_1$-penalised sparse precision matrix estimation
摘要：
各种$\ell_1$惩罚估计方法，如图形套索和CLIME广泛用于稀疏精度矩阵估计。许多这些方法已被证明是一致的，在各种定量假设的基础上真正的协方差矩阵。直觉上，这些条件与惩罚项将主导优化的情况有关。在本文中，我们探讨了基于$\ell_1$的方法对一类稀疏潜变量类模型的一致性，这类模型受到几种类型的应用的强烈激励。我们发现，所有基于$\ell_1$的方法显着失败的模型与变量之间的线性依赖关系。我们还研究了来自真实基因表达数据的模型的一致性，并注意到即使对于中等规模的基因网络，一致性所需的假设也从未成立，并且基于$\ell_1$的方法在实践中对于较大的网络也变得不可靠。
英文摘要：
Various $\ell_1$-penalised estimation methods such as graphical lasso and CLIME are widely used for sparse precision matrix estimation. Many of these methods have been shown to be consistent under various quantitative assumptions about the underlying true covariance matrix. Intuitively, these conditions are related to situations where the penalty term will dominate the optimisation. In this paper, we explore the consistency of $\ell_1$-based methods for a class of sparse latent variable -like models, which are strongly motivated by several types of applications. We show that all $\ell_1$-based methods fail dramatically for models with nearly linear dependencies between the variables. We also study the consistency on models derived from real gene expression data and note that the assumptions needed for consistency never hold even for modest sized gene networks and $\ell_1$-based methods also become unreliable in practice for larger networks.
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