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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.

Tony Cai develops methods and theory for high-dimensional statistics, statistical decision theory and statistical machine learning. His research includes functional data analysis and applications to genomics.^[ref]

With Weidong Liu and Xi Luo, he introduced CLIME, estimating a sparse precision matrix through linear programming. Their results bound estimation error under sparsity and tail assumptions. The method makes a useful distinction between an optimization constraint and a theorem: positive definiteness of the final estimate is established under conditions, rather than imposed as a constraint in the columnwise programs.^[ref]

Weidong Liu works on high-dimensional statistics. Shanghai Jiao Tong University's institutional profile identifies this researcher and his focus on high-dimensional statistical inference.^[ref]

His CLIME paper with Tony Cai and Xi Luo estimates a sparse precision matrix even when direct inversion of the sample covariance matrix is unsuitable. It controls $\Vert \hat{\Sigma }\Omega -I{\Vert }_{\max }$ while minimizing an entrywise absolute-value objective, then uses a symmetrization rule. Statistical accuracy requires assumptions on sparsity and the distribution's tails.^[ref]

Xi Luo develops biostatistics and machine learning methods for biomedical data. His institutional profile identifies applications in neuroscience, genomics and disease research.^[ref]

With Tony Cai and Weidong Liu, he developed CLIME for sparse precision matrix estimation. The method constrains the error of an approximate inverse of the sample covariance matrix and minimizes an entrywise absolute-value objective. Its computation separates into columnwise linear programs.^[ref]