dot-woodbury_redecompose_weighted: Per-Iteration Weighted Woodbury Redecomposition

In lgspline: Lagrangian Multiplier Smoothing Splines for Smooth Function Estimation

Per-Iteration Weighted Woodbury Redecomposition

Description

Weighted analogue of .woodbury_decompose_V(). At each GLM iteration, this helper rebuilds the current correlated information, absorbs the block-diagonal part into \mathbf{G}_{\mathrm{on}}^{-1}, and refactors the cross-partition remainder as \mathbf{E}\mathbf{J}\mathbf{E}^{\top}.

Usage

.woodbury_redecompose_weighted(
  Delta_V,
  X_block,
  DV_X,
  W,
  X,
  K,
  p_expansions,
  X_gram_weighted,
  Lambda,
  schur_corrections,
  unique_penalty_per_partition,
  L_partition_list,
  parallel_eigen,
  cl,
  chunk_size,
  num_chunks,
  rem_chunks,
  rank_threshold_fraction = 2/3,
  family
)

Arguments

Delta_V	Fixed N \times N matrix \mathbf{V}^{-1} - \mathbf{I}.
X_block	Full N \times P block-diagonal design matrix.
DV_X	Precomputed N \times P product of (\mathbf{V}^{-1} - \mathbf{I})\mathbf{X}.
W	Length-N vector of current GLM working weights.
X	List of partition-specific design matrices.
K, p_expansions	Integer dimensions.
X_gram_weighted	List of K+1 weighted Gram matrices \mathbf{X}_k^{\top}\mathrm{diag}(\mathbf{W}_k)\mathbf{X}_k.
Lambda	Shared p \times p penalty matrix.
schur_corrections	List of K+1 Schur correction matrices.
unique_penalty_per_partition	Logical.
L_partition_list	Partition-specific penalty matrices.
parallel_eigen	Logical; parallel eigendecomposition.
cl, chunk_size, num_chunks, rem_chunks	Parallel parameters.
rank_threshold_fraction	Numeric; Woodbury threshold (default 2/3).
family	GLM family object.

Details

It is standalone in the sense that it returns the full Woodbury state for the current working weights: corrected block-diagonal factors, the low-rank basis, and the small inverse (\mathbf{J}^{-1} + \mathbf{E}^{\top}\mathbf{G}_{\mathrm{on}}\mathbf{E})^{-1}.

Value

A list with the same structure as .woodbury_decompose_V:

use_woodbury

Logical; FALSE if rank exceeds threshold.

Ghalf_corrected

List of corrected \mathbf{G}_{\mathrm{on},k}^{1/2}.

GhalfInv_corrected

List of corrected \mathbf{G}_{\mathrm{on},k}^{-1/2}.

G_corrected

List of corrected \mathbf{G}_{\mathrm{on},k}.

E

Off-diagonal low-rank factor (P \times r), denoted by \mathbf{E} in the supplement.

J

Length-r sign vector, i.e.\ the diagonal of \mathbf{J}.

r

Integer effective rank.

inner_inv

Precomputed r \times r Woodbury inner inverse; in supplement notation, (\mathbf{J}^{-1} + \mathbf{E}^{\top}\mathbf{G}_{\mathrm{on}} \mathbf{E})^{-1}.

GL

List of per-partition matrices corresponding to \mathbf{G}_{\mathrm{on},k}\mathbf{E}[rows_k,] in supplement notation.

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