LogDetDiv

The log-det divergence of a pair of \(n \times n\) Hermitian positive-definite matrices \(A\) and \(B\) is

\[D_{ld}(A,B) = \mbox{tr}(A B^{-1}) -\log(\mbox{det}(A B^{-1})) - n,\]

which can be greatly simplified using the Cholesky factors of \(A\) and \(B\). In particular, if we set \(Z = L_B^{-1} L_A\), where \(A=L_A L_A^H\) and \(B=L_B L_B^H\) are Cholesky factorizations, then

\[D_{ld}(A,B) = \| Z \|_F^2 - 2 \log(\mbox{det}(Z)) - n.\]

Implementation

Example driver

C++ API

Base<F> LogDetDiv(UpperOrLower uplo, const Matrix<F> &A, const Matrix<F> &B)

Base<F> LogDetDiv(UpperOrLower uplo, const ElementalMatrix<F> &A, const ElementalMatrix<F> &B)

C API

ElError ElLogDetDiv_s(ElUpperOrLower uplo, ElConstMatrix_s A, ElConstMatrix_s B)

ElError ElLogDetDiv_d(ElUpperOrLower uplo, ElConstMatrix_d A, ElConstMatrix_d B)

ElError ElLogDetDiv_c(ElUpperOrLower uplo, ElConstMatrix_c A, ElConstMatrix_c B)

ElError ElLogDetDiv_z(ElUpperOrLower uplo, ElConstMatrix_z A, ElConstMatrix_z B)

ElError ElLogDetDivDist_s(ElUpperOrLower uplo, ElConstDistMatrix_s A, ElConstDistMatrix_s B)

ElError ElLogDetDivDist_d(ElUpperOrLower uplo, ElConstDistMatrix_d A, ElConstDistMatrix_d B)

ElError ElLogDetDivDist_c(ElUpperOrLower uplo, ElConstDistMatrix_c A, ElConstDistMatrix_c B)

ElError ElLogDetDivDist_z(ElUpperOrLower uplo, ElConstDistMatrix_z A, ElConstDistMatrix_z B)