FGfun: The Objective function and its gradient

In TRES: Tensor Regression with Envelope Structure

Description

Calculates the objective function and its gradient for estimating the M-envelope of span(U), where M is positive definite and U is positive semi-definite.

Usage

FGfun(Gamma, M, U)

Arguments

Gamma	Γ matrix in the envelope objective function. A p-by-u matrix.
M	The p-by-p positive definite matrix M in the envelope objective function.
U	The p-by-p positive semi-definite matrix U in the envelope objective function.

Details

The generic objective function F(Γ) and its gradient G(Γ) are listed below for estimating M-envelope of span(U). For the detailed description, see Cook, R. D., & Zhang, X. (2016).

F(Γ)=\log|Γ^T M Γ|+\log| Γ^T(M+U)^{-1}Γ|

G(Γ) = dF/d Γ = 2 M Γ (Γ^T M Γ)^{-1} + 2 (M + U)^{-1} Γ (Γ^T (M + U)^{-1} Γ)^{-1}

Value

F	The value of the objective function at Gamma.
G	The value of the gradient function at Gamma.

References

Cook, R.D. and Zhang, X., 2016. Algorithms for envelope estimation. Journal of Computational and Graphical Statistics, 25(1), pp.284-300.

TRES documentation built on Oct. 20, 2021, 9:06 a.m.