Elastic net¶

The elastic net (EN) seeks an approximate solution to a (usually undetermined) system of equations which penalizes both the one and two norms of the solution:

\[\min_x \| b - A x \|_2^2 + \lambda_1 \| x \|_1 + \lambda_2 \| x \|_2^2.\]

One could perhaps motivate the elastic net by recognizing that an LQ decomposition of an underdetermined systems of equations finds an \(x\) which exactly satisfies \(A x = b\) and simultaneously minimizes \(\| x \|_2\). From this perspective, for underdetermined systems, the elastic net trades the exact satisfaction of \(A x = b\), which would be provided via an LQ decomposition, for the hope of a sparse \(x\) via the addition of a penalty on \(\| x \|_1\).

Real instances of the problem are expressable as an (affine) quadratic program in a manner which is nearly identical to that of basis pursuit denoising (BPDN): by splitting \(x\) into its positive and negative components, say \(x = u - v\), and introducing the residual \(r = b - A x\), one arrives at

\[\begin{split}& \min_{u,v,r} r^T r + \lambda_2 (u^T u + v^T v) + \lambda_1 1^T (u + v) \\ & \text{s.t. } \begin{pmatrix} A & -A \end{pmatrix} \begin{pmatrix} u \\ v \end{pmatrix} + r = b,\; u,v \ge 0.\end{split}\]

By default, Elemental solves this equation using a Mehrotra Predictor-Corrector primal-dual Interior Point Method.

Python API¶

EN(A, b, lambda1, lambda2[, ctrl=None])¶

Parameters

A – dense or sparse, sequential or distributed matrix
b – dense right-hand side vector (with type compatible to A)
lambda1 – penalty on the one-norm of the solution
lambda2 – penalty on the square of the two-norm of the solution
ctrl – (optional) QPAffineCtrl instance

Return type

dense solution vector (with type matching that of b)

C++ API¶

void EN(const Matrix<Real> &A, const Matrix<Real> &b, Real lambda1, Real lambda2, Matrix<Real> &x, const qp::affine::Ctrl<Real> &ctrl = qp::affine::Ctrl<Real>())¶

void EN(const ElementalMatrix<Real> &A, const ElementalMatrix<Real> &b, Real lambda1, Real lambda2, ElementalMatrix<Real> &x, const qp::affine::Ctrl<Real> &ctrl = qp::affine::Ctrl<Real>())¶

void EN(const SparseMatrix<Real> &A, const Matrix<Real> &b, Real lambda1, Real lambda2, Matrix<Real> &x, const qp::affine::Ctrl<Real> &ctrl = qp::affine::Ctrl<Real>())¶

void EN(const DistSparseMatrix<Real> &A, const DistMultiVec<Real> &b, Real lambda1, Real lambda2, DistMultiVec<Real> &x, const qp::affine::Ctrl<Real> &ctrl = qp::affine::Ctrl<Real>())¶

C API¶

Single-precision¶

ElError ElEN_s(ElConstMatrix_s A, ElConstMatrix_s b, float lambda1, float lambda2, ElMatrix_s x)¶

ElError ElENDist_s(ElConstDistMatrix_s A, ElConstDistMatrix_s b, float lambda1, float lambda2, ElDistMatrix_s x)¶

ElError ElENSparse_s(ElConstSparseMatrix_s A, ElConstMatrix_s b, float lambda1, float lambda2, ElMatrix_s x)¶

ElError ElENDistSparse_s(ElConstDistSparseMatrix_s A, ElConstDistMultiVec_s b, float lambda1, float lambda2, ElDistMultiVec_s x)¶

Double-precision¶

ElError ElEN_d(ElConstMatrix_d A, ElConstMatrix_d b, double lambda1, double lambda2, ElMatrix_d x)¶

ElError ElENDist_d(ElConstDistMatrix_d A, ElConstDistMatrix_d b, double lambda1, double lambda2, ElDistMatrix_d x)¶

ElError ElENSparse_d(ElConstSparseMatrix_d A, ElConstMatrix_d b, double lambda1, double lambda2, ElMatrix_d x)¶

ElError ElENDistSparse_d(ElConstDistSparseMatrix_d A, ElConstDistMultiVec_d b, double lambda1, double lambda2, ElDistMultiVec_d x)¶

Expert interface¶

Single-precision¶

ElError ElENX_s(ElConstMatrix_s A, ElConstMatrix_s b, float lambda1, float lambda2, ElMatrix_s x, ElQPAffineCtrl_s ctrl)¶

ElError ElENXDist_s(ElConstDistMatrix_s A, ElConstDistMatrix_s b, float lambda1, float lambda2, ElDistMatrix_s x, ElQPAffineCtrl_s ctrl)¶

ElError ElENXSparse_s(ElConstSparseMatrix_s A, ElConstMatrix_s b, float lambda1, float lambda2, ElMatrix_s x, ElQPAffineCtrl_s ctrl)¶

ElError ElENXDistSparse_s(ElConstDistSparseMatrix_s A, ElConstDistMultiVec_s b, float lambda1, float lambda2, ElDistMultiVec_s x, ElQPAffineCtrl_s ctrl)¶

Double-precision¶

ElError ElENX_d(ElConstMatrix_d A, ElConstMatrix_d b, double lambda1, double lambda2, ElMatrix_d x, ElQPAffineCtrl_d ctrl)¶

ElError ElENXDist_d(ElConstDistMatrix_d A, ElConstDistMatrix_d b, double lambda1, double lambda2, ElDistMatrix_d x, ElQPAffineCtrl_d ctrl)¶

ElError ElENXSparse_d(ElConstSparseMatrix_d A, ElConstMatrix_d b, double lambda1, double lambda2, ElMatrix_d x, ElQPAffineCtrl_d ctrl)¶

ElError ElENXDistSparse_d(ElConstDistSparseMatrix_d A, ElConstDistMultiVec_d b, double lambda1, double lambda2, ElDistMultiVec_d x, ElQPAffineCtrl_d ctrl)¶