dot-get_B_gee_woodbury: Path 1a-Woodbury: Gaussian GEE with Woodbury Acceleration

In lgspline: Lagrangian Multiplier Smoothing Splines for Smooth Function Estimation

Path 1a-Woodbury: Gaussian GEE with Woodbury Acceleration

Description

Gaussian correlated solve using the Woodbury factorization from the supplement. This replaces the dense GEE path when \mathbf{G}_{\mathrm{off}}^{-1} = \mathbf{E}\mathbf{J}\mathbf{E}^{\top} has low rank and computes the constrained estimate through the same transformed OLS projection as the independent case, with \mathbf{G}^{1/2} = \mathbf{G}_{\mathrm{on}}^{1/2}\mathbf{F}^{1/2}.

Usage

.get_B_gee_woodbury(
  X,
  y,
  K,
  p_expansions,
  VhalfInv_perm,
  order_list,
  A,
  R_constraints,
  constraint_value_vectors,
  family,
  return_G_getB,
  quadprog,
  qp_Amat,
  qp_bvec,
  qp_meq,
  qp_score_function,
  observation_weights,
  wb_decomp,
  wb_sqrt,
  Lambda_block,
  parallel_aga,
  parallel_matmult,
  cl,
  chunk_size,
  num_chunks,
  rem_chunks,
  qp_global,
  tol,
  parallel_qr = FALSE,
  initial_active_ineq = integer(0),
  ...
)

Arguments

X, y	Lists of partition-specific design matrices and responses.
K, p_expansions	Integer dimensions.
VhalfInv_perm	\mathbf{V}^{-1/2} permuted to partition ordering.
order_list	Partition-to-data index mapping.
A	Constraint matrix (P \times R).
R_constraints	Number of columns of A.
constraint_value_vectors	Constraint RHS list.
family	GLM family object.
return_G_getB	Logical; return covariance components.
quadprog	Logical; apply QP refinement.
qp_Amat, qp_bvec, qp_meq	QP constraint specification.
qp_score_function	Score function for QP step.
observation_weights	Observation weights.
wb_decomp	Output of .woodbury_decompose_V.
wb_sqrt	Output of .woodbury_halfsqrt_components.
Lambda_block	Full block-diagonal penalty matrix.
parallel_aga, parallel_matmult	Logical flags.
cl, chunk_size, num_chunks, rem_chunks	Parallel parameters.
qp_global	Logical; TRUE when inequality constraints couple partitions and therefore require the global equality bridge inside the active-set refinement.
tol	Numeric tolerance used by the Woodbury active-set refinement.
...	Passed to sub-functions.

Details

This function is standalone for the Gaussian Woodbury path: it takes the decomposition from .woodbury_decompose_V(), the square-root state from .woodbury_halfsqrt_components(), performs the constrained solve, and returns the same object structure as the dense Gaussian GEE solver.

Cost is O(Kp^3 + Pr^2) compared to O(P^3) for the dense Path 1a.

Value

Same structure as .get_B_gee_gaussian.

lgspline documentation built on Aug. 5, 2026, 1:10 a.m.