dot-woodbury_decompose_V: Woodbury Decomposition of Correlation Structure
In lgspline: Lagrangian Multiplier Smoothing Splines for Smooth Function Estimation

.woodbury_decompose_V

R Documentation

Woodbury Decomposition of Correlation Structure

Description

Builds the Woodbury factors for the correlated solver. The function decomposes \mathbf{G}^{-1} = \mathbf{G}_{\mathrm{on}}^{-1} + \mathbf{G}_{\mathrm{off}}^{-1} by absorbing the block-diagonal part of \mathbf{X}^{\top}\mathbf{V}^{-1}\mathbf{X} + \boldsymbol{\Lambda} into \mathbf{G}_{\mathrm{on}}^{-1}, then factoring the cross-partition part as \mathbf{G}_{\mathrm{off}}^{-1} = \mathbf{E}\mathbf{J}\mathbf{E}^{\top}.

Usage

.woodbury_decompose_V(
  VhalfInv_perm,
  X,
  K,
  p_expansions,
  Lambda,
  Lambda_block,
  unique_penalty_per_partition,
  L_partition_list,
  order_list,
  parallel_eigen,
  cl,
  chunk_size,
  num_chunks,
  rem_chunks,
  rank_threshold_fraction = 2/3,
  family
)

Arguments

`VhalfInv_perm`	`\mathbf{V}^{-1/2}` permuted to partition ordering.
`X`	List of partition-specific design matrices.
`K`, `p_expansions`	Integer dimensions.
`Lambda`, `Lambda_block`	Shared and full block-diagonal penalty matrices.
`unique_penalty_per_partition`	Logical.
`L_partition_list`	Partition-specific penalty matrices.
`order_list`	Partition-to-data index mapping.
`parallel_eigen`	Logical; parallel eigendecomposition.
`cl`, `chunk_size`, `num_chunks`, `rem_chunks`	Parallel parameters.
`rank_threshold_fraction`	Numeric; Woodbury is used only when `r < P \times` this fraction (default 2/3).
`family`	GLM family object.

Details

The output is exactly the information needed by .woodbury_halfsqrt_components() and .lagrangian_project_woodbury(): corrected block-diagonal \mathbf{G}_{\mathrm{on}}, the low-rank factor \mathbf{E}, the sign matrix diagonal \mathbf{J}, and the small inverse (\mathbf{J}^{-1} + \mathbf{E}^{\top}\mathbf{G}_{\mathrm{on}}\mathbf{E})^{-1}.

When the cross-partition rank is zero or too large to be worthwhile, the helper returns use_woodbury = FALSE so the caller can use the dense correlated path instead.

Value

A list with components:

use_woodbury: Logical; FALSE triggers dense Path 1 fallback.
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) from the supplement factorization \mathbf{G}_{\mathrm{off}}^{-1} = \mathbf{E}\mathbf{J}\mathbf{E}^{\top}.
J: Length-r vector giving the diagonal of \mathbf{J}.
r: Integer effective rank.
inner_inv: Precomputed r \times r Woodbury inner inverse. In supplement notation this is (\mathbf{J}^{-1} + \mathbf{E}^{\top}\mathbf{G}_{\mathrm{on}} \mathbf{E})^{-1}.
GL: List of K+1 matrices storing \mathbf{G}_{\mathrm{on},k}\mathbf{E}[rows_k,] in supplement notation, each p \times r.