标题： 图解套索解决方案的存在性 显示英文标题

标题： Existence of the solution to the graphical lasso

摘要： 图形lasso（glasso）是一种针对高斯精度矩阵的$l_1$惩罚似然估计器。 glasso的一个好处是，即使样本协方差矩阵不是正定的而是半正定的，它仍然存在。 本文收集了关于glasso在对精度矩阵的所有元素应用惩罚以及仅对非对角线元素应用惩罚时存在性的若干结果。 为这些结果提供了新的证明，这些证明揭示了$l_1$惩罚如何实现这些存在性属性。 这些证明可以推广到更大一类惩罚函数，从而可以轻松确定对于半正定样本协方差的新惩罚似然估计是否存在。 显示英文摘要

摘要： The graphical lasso (glasso) is an $l_1$ penalised likelihood estimator for a Gaussian precision matrix. A benefit of the glasso is that it exists even when the sample covariance matrix is not positive definite but only positive semidefinite. This note collects a number of results concerning the existence of the glasso both when the penalty is applied to all entries of the precision matrix and when the penalty is only applied to the off-diagonals. New proofs are provided for these results which give insight into how the $l_1$ penalty achieves these existence properties. These proofs extend to a much larger class of penalty functions allowing one to easily determine if new penalised likelihood estimates exist for positive semidefinite sample covariance.