TRES (version 1.1.3)

manifold1D: Estimate the envelope subspace (ManifoldOptim 1D)

Description

The 1D algorithm (Cook and Zhang 2016) implemented with Riemannian manifold optimization from R package ManifoldOptim.

Usage

manifold1D(M, U, u, ...)

Arguments

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.

u

An integer between 0 and $n$ representing the envelope dimension.

...

Additional user-defined arguments:

• maxiter: The maximal number of iterations.

• tol: The tolerance used to assess convergence. See Huang et al. (2018) for details on how this is used.

• method: The name of optimization method supported by R package ManifoldOptim.

  • "LRBFGS": Limited-memory RBFGS

  • "LRTRSR1": Limited-memory RTRSR1

  • "RBFGS": Riemannian BFGS

  • "RBroydenFamily": Riemannian Broyden family

  • "RCG": Riemannian conjugate gradients

  • "RNewton": Riemannian line-search Newton

  • "RSD": Riemannian steepest descent

  • "RTRNewton": Riemannian trust-region Newton

  • "RTRSD": Riemannian trust-region steepest descent

  • "RTRSR1": Riemannian trust-region symmetric rank-one update

  • "RWRBFGS": Riemannian BFGS

• check: Logical value. Should internal manifold object check inputs and print summary message before optimization.

The default values are: maxiter = 500; tol = 1e-08; method = "RCG"; check = FALSE.

Value

Return the estimated orthogonal basis of the envelope subspace.

Details

Estimate M-envelope of span(U). The dimension of the envelope is u.

References

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

Huang, W., Absil, P.A., Gallivan, K.A. and Hand, P., 2018. ROPTLIB: an object-oriented C++ library for optimization on Riemannian manifolds. ACM Transactions on Mathematical Software (TOMS), 44(4), pp.1-21.

See Also

MenvU_sim, subspace

Examples

# NOT RUN {
## Simulate two matrices M and U with an envelope structure
data <- MenvU_sim(p = 20, u = 5, wishart = TRUE, n = 200)
M <- data$M
U <- data$U
G <- data$Gamma
Gamma_1D <- manifold1D(M, U, u = 5)
subspace(Gamma_1D, G)

# }