Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 205 / 2

The Gaussian maximum-likelihood covariance estimate isFor each $i,k$,so $\Vert AB{\Vert }_{\infty }\le \Vert A{\Vert }_{\infty }\Vert B{\Vert }_{L^1}$. If $A$ is square and symmetric, its maximum row sum equals its maximum column sum, and the same calculation gives

Both the objective $\Vert \Omega {\Vert }_1={\sum }_j\Vert {\Omega }_j{\Vert }_1$ and the constraints separate by columns. Replacing one column of a global minimizer by a better feasible column would improve the global objective. Therefore each ${\hat{\Omega }}_j$ minimizesMoreover,Thus ${\Omega }_j^0$ is feasible andFor every column,Finally $\hat{\Omega }-{\Omega }^0={\Omega }^0{\Sigma }^0(\hat{\Omega }-{\Omega }^0)$, and symmetry of ${\Omega }^0$ givesThis is the basic error bound for the CLIME precision-matrix estimator.