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The CLIME precision-matrix estimator estimates a sparse matrix representing the precision matrix from a sample covariance matrix $S$. An initial estimate $B$ solvesThese are the entrywise matrix L1 norm and entrywise maximum norm, not the induced column-sum and row-sum norms. Writing $b_j$ for column $j$ and $e_j$ for the corresponding basis vector gives separate problems $\min \Vert b_j{\Vert }_1$ subject to $\Vert S{b}_j-e_j{\Vert }_{\infty }\le \lambda$, each a linear programming problem.^[ref]

The column solutions need not form a symmetric matrix. The original construction selects the smaller-magnitude entry from each pair $B_{ij},B_{ji}$ and copies it to both positions, using a common tie rule. Error analysis requires that the true precision matrix be feasible for the chosen tolerance.^[ref]

Positive definiteness is not an optimization constraint here. For example, with $S=I$ and $\lambda =1$, the zero matrix is feasible and minimizes the objective. A positive-definite conclusion therefore needs further assumptions and a statistical theorem, rather than following from the program alone.

The Graphical Lasso estimates a sparse matrix for the precision matrix by minimizing a penalized Gaussian objective over positive-definite matrices. With a sample covariance matrix $S$ and $\lambda >0$, the all-entry penalty convention isThe entrywise matrix L1 norm encourages zero entries, while the domain enforces positive definiteness. In a Gaussian graphical model, zero off-diagonal precision matrix entries encode conditional independence given the other coordinates.^[ref]

The penalty convention matters. Writing $W={\hat{\Omega }}^{-1}$, the Graphical-Lasso Karush-Kuhn-Tucker conditions give $W_{ii}=S_{ii}+\lambda$ when diagonal entries are penalized, but $W_{ii}=S_{ii}$ when they are excluded.^[ref] For example, diagonal $S=\mathrm{diag}(1,4)$ and $\lambda =1$ give $\hat{\Omega }=\mathrm{diag}(1/2,1/5)$ under the all-entry convention.

The Graphical Lasso column regression relates its matrix optimization to Lasso subproblems. This is different from independently penalizing a raw covariance inverse, and the CLIME precision-matrix estimator uses a different constrained objective.

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.