= CLIME precision-matrix estimator
{c}
{title2=$\widehat\Omega_{\mathrm{CLIME}}$}
The <CLIME precision-matrix estimator> estimates a <sparse matrix> representing the <precision matrix> from a <sample covariance matrix> $S$. An initial estimate $B$ solves
$$
\min_B\sum_{i,j}|B_{ij}|\quad\text{subject to}\quad\max_{i,j}|(SB-I)_{ij}|\leq\lambda.
$$
These 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 $\min\|b_j\|_1$ subject to $\|Sb_j-e_j\|_\infty\leq\lambda$, each a <linear programming> problem.<A Constrained L1 Minimization Approach to Sparse Precision Matrix Estimation by Tony Cai and others>[Section 2, equations 1–3: norm definitions, constrained estimator and column decomposition]{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.<A Constrained L1 Minimization Approach to Sparse Precision Matrix Estimation by Tony Cai and others>[Section 2, symmetrization after equation 1 and feasibility-based analysis]{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.
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