Singular-value soft-thresholding

Overwrites \(A\) with \(U S_{\tau}(\Sigma) V^H\), where \(U \Sigma V^H\) is the singular-value decomposition of \(A\) upon input and \(S_{\tau}\) performs soft-thresholding with parameter \(\tau\). The return value is the rank of the soft-thresholded matrix.

Implementation

Standard algorithm

Runs the default SVT algorithm. In the sequential case, this is currently svt::Normal, and, in the parallel case, it is svt::Cross.

C++ API

Int SVT(Matrix<F> &A, Base<F> tau, bool relative = false)

Int SVT(AbstractDistMatrix<F> &A, Base<F> tau, bool relative = false)

C API

ElError ElSVT_s(ElMatrix_s A, float rho, bool relative)

ElError ElSVT_d(ElMatrix_d A, double rho, bool relative)

ElError ElSVT_c(ElMatrix_c A, float rho, bool relative)

ElError ElSVT_z(ElMatrix_z A, double rho, bool relative)

ElError ElSVTDist_s(ElDistMatrix_s A, float rho, bool relative)

ElError ElSVTDist_d(ElDistMatrix_d A, double rho, bool relative)

ElError ElSVTDist_c(ElDistMatrix_c A, float rho, bool relative)

ElError ElSVTDist_z(ElDistMatrix_z A, double rho, bool relative)

Approximate algorithm

Runs a faster (for small ranks), but less accurate, algorithm given an upper bound on the rank of the soft-thresholded matrix. The current implementation preprocesses via relaxedRank steps of (Businger-Golub) column-pivoted QR via the routine svt::PivotedQR.

C++ API

Int SVT(Matrix<F> &A, Base<F> tau, Int relaxedRank, bool relative = false)

Int SVT(AbstractDistMatrix<F> &A, Base<F> tau, Int relaxedRank, bool relative = false)

C API

TODO

Tall-skinny algorithm

Runs an SVT algorithm designed for tall-skinny matrices. The current implementation is based on TSQR factorization and is svt::TSQR.

C++ API

Int SVT(DistMatrix<F, U, STAR> &A, Base<F> tau, bool relative = false)

C API

TODO

namespace svt

Int svt::Normal(Matrix<F> &A, Base<F> tau, bool relative = false)

Int svt::Normal(AbstractDistMatrix<F> &A, Base<F> tau, bool relative = false)

Runs a standard SVD, soft-thresholds the singular values, and then reforms the matrix.

Int svt::Cross(Matrix<F> &A, Base<F> tau, bool relative = false)

Int svt::Cross(AbstractDistMatrix<F> &A, Base<F> tau, bool relative = false)

Forms the normal matrix, computes its Hermitian EVD, soft-thresholds the eigenvalues, and then reforms the matrix. Note that Elemental’s parallel Hermitian EVD is much faster than its parallel SVD; this is typically worth the loss of accuracy in the computed small (truncated) singular values and is therefore the default choice for parallel SVT.

Int svt::PivotedQR(Matrix<F> &A, Base<F> tau, Int numStepsQR, bool relative = false)

Int svt::PivotedQR(AbstractDistMatrix<F> &A, Base<F> tau, Int numStepsQR, bool relative = false)

Computes an approximate SVT by first approximating A as the rank-numSteps approximation produced by numSteps iterations of column-pivoted QR.

Int svt::TSQR(AbstractDistMatrix<F> &A, Base<F> tau, bool relative = false)

Since the majority of the work in a tall-skinny SVT will be in the initial QR factorization, this algorithm runs a TSQR factorization and then computes the SVT of the small R factor using a single process.