Least Absolute Value regression¶

Least Absolute Value (LAV) regression minimizes a residual \(b - A x\) in the one-norm, i.e.,

\[\min_x \| A x - b \|_1,\]

and is known to be the solution of a linear program when \(A\) and \(b\) are real. In particular, we may solve

\[\begin{split}\min_{x,u,v} \{\; 1^T \begin{pmatrix} u \\ v \end{pmatrix} \; | \; \begin{pmatrix} A & I & -I \end{pmatrix} \begin{pmatrix} x \\ u \\ v \end{pmatrix} = b \; \wedge \; u,v \ge 0 \; \}.\end{split}\]

By default, Elemental solves this linear program via a Mehrotra Predictor-Corrector primal-dual Interior Point Method.

Python API¶

LAV(A, b[, ctrl=None])¶

Parameters

A – dense or sparse, sequential or distributed matrix
b – dense right-hand side vector (with type compatible to A)
ctrl – (optional) LPAffineCtrl instance

Return type

dense solution vector (with type matching that of b)

C++ API¶

void LAV(const Matrix<Real> &A, const Matrix<Real> &b, Matrix<Real> &x, const lp::affine::Ctrl<Real> &ctrl = lp::affine::Ctrl<Real>())¶

void LAV(const ElementalMatrix<Real> &A, const ElementalMatrix<Real> &b, ElementalMatrix<Real> &x, const lp::affine::Ctrl<Real> &ctrl = lp::affine::Ctrl<Real>())¶

void LAV(const SparseMatrix<Real> &A, const Matrix<Real> &b, Matrix<Real> &x, const lp::affine::Ctrl<Real> &ctrl = lp::affine::Ctrl<Real>())¶

void LAV(const DistSparseMatrix<Real> &A, const DistMultiVec<Real> &b, DistMultiVec<Real> &x, const lp::affine::Ctrl<Real> &ctrl = lp::affine::Ctrl<Real>())¶

C API¶

Single-precision¶

ElError ElLAV_s(ElConstMatrix_s A, ElConstMatrix_s b, ElMatrix_s x)¶

ElError ElLAVDist_s(ElConstDistMatrix_s A, ElConstDistMatrix_s b, ElDistMatrix_s x)¶

ElError ElLAVSparse_s(ElConstSparseMatrix_s A, ElConstMatrix_s b, ElMatrix_s x)¶

ElError ElLAVDistSparse_s(ElConstDistSparseMatrix_s A, ElConstDistMultiVec_s b, ElDistMultiVec_s x)¶

Double-precision¶

ElError ElLAV_d(ElConstMatrix_d A, ElConstMatrix_d b, ElMatrix_d x)¶

ElError ElLAVDist_d(ElConstDistMatrix_d A, ElConstDistMatrix_d b, ElDistMatrix_d x)¶

ElError ElLAVSparse_d(ElConstSparseMatrix_d A, ElConstMatrix_d b, ElMatrix_d x)¶

ElError ElLAVDistSparse_d(ElConstDistSparseMatrix_d A, ElConstDistMultiVec_d b, ElDistMultiVec_d x)¶

Expert interface¶

Single-precision¶

ElError ElLAVX_s(ElConstMatrix_s A, ElConstMatrix_s b, ElMatrix_s x, ElLPAffineCtrl_s ctrl)¶

ElError ElLAVXDist_s(ElConstDistMatrix_s A, ElConstDistMatrix_s b, ElDistMatrix_s x, ElLPAffineCtrl_s ctrl)¶

ElError ElLAVXSparse_s(ElConstSparseMatrix_s A, ElConstMatrix_s b, ElMatrix_s x, ElLPAffineCtrl_s ctrl)¶

ElError ElLAVXDistSparse_s(ElConstDistSparseMatrix_s A, ElConstDistMultiVec_s b, ElDistMultiVec_s x, ElLPAffineCtrl_s ctrl)¶

Double-precision¶

ElError ElLAVX_d(ElConstMatrix_d A, ElConstMatrix_d b, ElMatrix_d x, ElLPAffineCtrl_d ctrl)¶

ElError ElLAVXDist_d(ElConstDistMatrix_d A, ElConstDistMatrix_d b, ElDistMatrix_d x, ElLPAffineCtrl_d ctrl)¶

ElError ElLAVXSparse_d(ElConstSparseMatrix_d A, ElConstMatrix_d b, ElMatrix_d x, ElLPAffineCtrl_d ctrl)¶

ElError ElLAVXDistSparse_d(ElConstDistSparseMatrix_d A, ElConstDistMultiVec_d b, ElDistMultiVec_d x, ElLPAffineCtrl_d ctrl)¶