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)