# Sparse inverse covariance selection¶

The following routines attempt to find a sparse inverse covariance matrix which could generate the given observations. This search is performed by attempting to solve the program

$\text{min trace}(S Z) - \text{log}\;\text{det}\;Z + \lambda \|\text{vec}(Z)\|_1$

using the Alternating Direction Method of Multipliers.

The following functions were inspired by a simple ADMM solver due to Boyd et al. Elemental’s implementations make use of parallel (dense) linear algebra (including PMRRR for the symmetric tridiagonal eigensolver).

## C++ API¶

Int SparseInvCov(const Matrix<F> &D, Base<F> lambda, Matrix<F> &Z, Base<F> rho = 1., Base<F> alpha = 1.2, Int maxIter = 500, Base<F> absTol = 1e-6, Base<F> relTol = 1e-4, bool progress = true)
Int SparseInvCov(const AbstractDistMatrix<F> &D, Base<F> lambda, AbstractDistMatrix<F> &Z, Base<F> rho = 1., Base<F> alpha = 1.2, Int maxIter = 500, Base<F> absTol = 1e-6, Base<F> relTol = 1e-4, bool progress = true)

Implementations

Example driver

Parameters Input/Output Explanation
D Input Observations
Z Output primal solution (second term)
lambda Input coefficient for vector-l1 penalty
rho Input augmented-Lagrangian parameter
alpha Input over-relaxation parameter (usually in $$[1,1.8]$$)
maxIter Input maximum number of ADMM iterations
absTol Input absolute convergence tolerance
relTol Input relative convergence tolerance
progress Input display detailed progress information?

## C API¶

ElError ElSparseInvCov_s(ElConstMatrix_s D, float lambda, ElMatrix_s Z, ElInt* numIts)
ElError ElSparseInvCov_d(ElConstMatrix_d D, double lambda, ElMatrix_d Z, ElInt* numIts)
ElError ElSparseInvCov_c(ElConstMatrix_c D, float lambda, ElMatrix_c Z, ElInt* numIts)
ElError ElSparseInvCov_z(ElConstMatrix_z D, double lambda, ElMatrix_z Z, ElInt* numIts)
ElError ElSparseInvCovDist_s(ElConstDistMatrix_s D, float lambda, ElDistMatrix_s Z, ElInt* numIts)
ElError ElSparseInvCovDist_d(ElConstDistMatrix_d D, double lambda, ElDistMatrix_d Z, ElInt* numIts)
ElError ElSparseInvCovDist_c(ElConstDistMatrix_c D, float lambda, ElDistMatrix_c Z, ElInt* numIts)
ElError ElSparseInvCovDist_z(ElConstDistMatrix_z D, double lambda, ElDistMatrix_z Z, ElInt* numIts)