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Outer Product of Canonical Gradients

Source: R/opcg_wrap_cpp.R
opcg.Rd

This implements the Outer Product of Canonical Gradients (OPCG) in a forth coming paper Quach and Li (2021).

Usage

opcg(
  x,
  y,
  d,
  bw,
  lambda = 0,
  ytype = "continuous",
  method = "newton",
  parallelize = F,
  r_mat = NULL,
  control_list = list()
)

Arguments

x

a 'nxp' matrix of predictors;

y

a 'nxm' response

d

specified the reduced dimension

bw

the bandwidth parameter for the kernel; the default kernel is gaussian

ytype

specify the response as 'continuous', 'multinomial', or 'ordinal'

method

"newton" or "cg" methods; for carrying out the optimization using the standard newton-raphson (i.e. Fisher Scoring) or using Congugate Gradients

parallelize

Default is False; to run in parallel, you will need to have foreach and some parallel backend loaded; parallelization is strongly recommended and encouraged.

control_list

a list of control parameters for the Newton-Raphson or Conjugate Gradient methods

  • opcg - A 'pxd' matrix that estimates a basis for the central subspace.

  • opcg_wls - A 'pxd' matrix that estimates a basis for the central subspace based on the initial value of the optimization problem; useful for examining bad starting values.

  • cand_mat - A list that contains both the candidate matrix for OPCG and for the initial value; this is used in other functions for order determination

  • gradients - The estimated local gradients; used in regularization of OPCG

  • weights - The kernel weights in the local-linear GLM.

Value

A 'pxd' matrix that estimates a basis for the central subspace based on the estimated local gradients

Details

The kernel for the local linear regression is fixed at a gaussian kernel.

For large 'p', we strongly recommend using the Conjugate Gradients implement, by setting method="cg". For method="cg", the hybrid conjugate gradient of Dai and Yuan is implemented, but only the armijo rule is implemented through backtracking, like in Bertsekas' "Convex Optimization Algorithms". A weak Wolfe condition can also be enforced by adding setting c_wolfe > 0 in the control_list, but since c_wolfe is usually set to 0.1 (Wikipedia) and this drastically slows down the algorithm relative to newton for small to moderate p, we leave the default as not enforcing a Wolfe condition, since we assume that our link function gives us a close enough initial point that local convergence is satisfactory. Should the initial values be suspect, then maybe enforcing the Wolfe condition is a reasonable trade-off.