Convex optimization¶
LogBarrier¶
Uses a careful calculation of the log of the determinant in order to return the log barrier of a Hermitian positive-definite matrix A, \(-\log(\mbox{det}(A))\).
- typename Base<F>::type LogBarrier(UpperOrLower uplo, const Matrix<F>& A)¶
- typename Base<F>::type LogBarrier(UpperOrLower uplo, const DistMatrix<F>& A)¶
- typename Base<F>::type LogBarrier(UpperOrLower uplo, Matrix<F>& A, bool canOverwrite=false )¶
- typename Base<F>::type LogBarrier(UpperOrLower uplo, DistMatrix<F>& A, bool canOverwrite=false )¶
LogDetDivergence¶
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
\[D_{ld}(A,B) = \| Z \|_F^2 - 2 \log(\mbox{det}(Z)) - n.\]
- typename Base<F>::type LogDetDivergence(UpperOrLower uplo, const Matrix<F>& A, const Matrix<F>& B)¶
- typename Base<F>::type LogDetDivergence(UpperOrLower uplo, const DistMatrix<F>& A, const DistMatrix<F>& B)¶