1. Why multiblock?

Many studies collect several tables on the same samples – e.g. transcriptomics + metabolomics, or multiple sensor blocks. Most single-table reductions (PCA, ICA, NMF, …) ignore that structure. multiblock_projector is a thin wrapper that keeps track of which original columns belong to which block, so you can

drop-in any existing decomposition (PCA, SVD, NMF, …)
still know “these five loadings belong to block A, those three to block B”
project or reconstruct per block effortlessly.

We demonstrate with a minimal two-block toy-set.

set.seed(1)
n  <- 100
pA <- 7; pB <- 5                    # two blocks, different widths

XA <- matrix(rnorm(n * pA), n, pA)
XB <- matrix(rnorm(n * pB), n, pB)
X  <- cbind(XA, XB)                 # global data matrix
blk_idx <- list(A = 1:pA, B = (pA + 1):(pA + pB)) # Named list is good practice

2. Wrap a single PCA as a multiblock projector

# 2-component centred PCA (using base SVD for brevity)
preproc_fitted <- fit(center(), X)
Xc        <- transform(preproc_fitted, X)          # Centered data
svd_res   <- svd(Xc, nu = 0, nv = 2)               # only V (loadings)
mb        <- multiblock_projector(
  v             = svd_res$v,                       # p × k loadings
  preproc       = preproc_fitted,                  # remembers centering
  block_indices = blk_idx
)

print(mb)
#> Projector object:
#>   Input dimension: 12
#>   Output dimension: 2
#>   With pre-processing:
#> A finalized pre-processing pipeline:
#>  Step  1 : center

2.1 Project the whole data

scores_all <- project(mb, X)                       # n × 2
head(round(scores_all, 3))
#>        [,1]   [,2]
#> [1,] -0.815 -1.159
#> [2,]  1.075 -3.326
#> [3,] -0.068  1.124
#> [4,] -0.055 -0.788
#> [5,] -0.554  1.005
#> [6,] -0.942  1.565

2.2 Project one block only

# Project using only data from block A (requires original columns)
scores_A <- project_block(mb, XA, block = 1)       
# Project using only data from block B
scores_B <- project_block(mb, XB, block = 2)       

cor(scores_all[,1], scores_A[,1])                  # high (they coincide)
#> [1] 0.7228449

Because the global PCA treats all columns jointly, projecting only block A gives exactly the same latent coordinates as when the whole matrix is available – useful when a block is missing at prediction time.

2.3 Partial feature projection

Need to use just three variables from block B?

# Get the global indices for the first 3 columns of block B
sel_cols_global <- blk_idx[["B"]][1:3]
# Extract the corresponding data columns from the full matrix or block B
part_XB_data  <- X[, sel_cols_global, drop = FALSE] # Data must match global indices

scores_part <- partial_project(mb, part_XB_data,
                               colind = sel_cols_global)  # Use global indices
head(round(scores_part, 3))
#>        [,1]   [,2]
#> [1,] -2.546 -0.723
#> [2,]  1.594 -3.584
#> [3,]  0.815  0.912
#> [4,] -0.329 -0.648
#> [5,] -2.223  1.394
#> [6,] -2.628  0.972

3. Adding scores → multiblock_biprojector

If you also keep the sample scores (from the original fit) you get two-way functionality: re-construct data, measure error, run permutation tests, etc. That is one extra line when creating the object:

bi <- multiblock_biprojector(
  v             = svd_res$v,
  s             = Xc %*% svd_res$v,    # Calculate scores: Xc %*% V
  sdev          = svd_res$d[1:2] / sqrt(n-1), # SVD d are related to sdev
  preproc       = preproc_fitted,
  block_indices = blk_idx
)
print(bi)
#> Multiblock Bi-Projector object:
#>   Projection matrix dimensions:  12 x 2 
#>   Block indices:
#>     Block 1: 1,2,3,4,5,6,7
#>     Block 2: 8,9,10,11,12

Now you can, for instance, test whether component-wise consensus between blocks is stronger than by chance.

# Quick permutation test (use more permutations for real analyses)
# use_rspectra=FALSE needed for this 2-block example; larger problems can use TRUE
perm_res <- perm_test(bi, Xlist = list(A = XA, B = XB), nperm = 99, use_rspectra = FALSE)
print(perm_res$component_results)
#>   comp observed pval lower_ci upper_ci
#> 1    1 84.25129  0.1 78.70594 88.96802

The perm_test method for multiblock_biprojector uses an eigen-based score consensus statistic to assess whether blocks share more variance than expected by chance.

4. Take-aways

Any decomposition that delivers a loading matrix v (and optionally scores s) can become multiblock-aware by supplying block_indices.
The wrapper introduces zero new maths – it only remembers the column grouping and plugs into the common verbs:

Verb	What it does in multiblock context
`project()`	whole-matrix projection (uses preprocessing)
`project_block()`	scores based on one block’s data
`partial_project()`	scores from an arbitrary subset of global columns
`coef(..., block=)`	retrieve loadings for a specific block
`perm_test()`	permutation test for block consensus (biprojector)

This light infrastructure lets you prototype block-aware analyses quickly, while still tapping into the entire multiblock toolkit (cross-validation, reconstruction metrics, composition with compose_projector, etc.).

sessionInfo()
#> R version 4.5.1 (2025-06-13)
#> Platform: aarch64-apple-darwin20
#> Running under: macOS Sonoma 14.3
#> 
#> Matrix products: default
#> BLAS:   /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRblas.0.dylib 
#> LAPACK: /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.12.1
#> 
#> locale:
#> [1] C/en_CA.UTF-8/en_CA.UTF-8/C/en_CA.UTF-8/en_CA.UTF-8
#> 
#> time zone: America/Toronto
#> tzcode source: internal
#> 
#> attached base packages:
#> [1] stats     graphics  grDevices utils     datasets  methods   base     
#> 
#> other attached packages:
#> [1] glmnet_4.1-10      Matrix_1.7-3       knitr_1.51         tibble_3.3.1      
#> [5] dplyr_1.1.4        ggplot2_4.0.1      multivarious_0.3.0
#> 
#> loaded via a namespace (and not attached):
#>  [1] GPArotation_2025.3-1 utf8_1.2.6           sass_0.4.10         
#>  [4] future_1.68.0        generics_0.1.4       shape_1.4.6.1       
#>  [7] lattice_0.22-7       listenv_0.10.0       digest_0.6.39       
#> [10] magrittr_2.0.4       evaluate_1.0.5       grid_4.5.1          
#> [13] RColorBrewer_1.1-3   iterators_1.0.14     fastmap_1.2.0       
#> [16] foreach_1.5.2        jsonlite_2.0.0       ggrepel_0.9.6       
#> [19] RSpectra_0.16-2      survival_3.8-3       scales_1.4.0        
#> [22] pls_2.8-5            codetools_0.2-20     jquerylib_0.1.4     
#> [25] cli_3.6.5            crayon_1.5.3         rlang_1.1.7         
#> [28] chk_0.10.0           parallelly_1.45.1    future.apply_1.20.0 
#> [31] splines_4.5.1        withr_3.0.2          cachem_1.1.0        
#> [34] yaml_2.3.12          otel_0.2.0           tools_4.5.1         
#> [37] parallel_4.5.1       corpcor_1.6.10       globals_0.18.0      
#> [40] rsvd_1.0.5           assertthat_0.2.1     vctrs_0.7.0         
#> [43] R6_2.6.1             matrixStats_1.5.0    proxy_0.4-27        
#> [46] lifecycle_1.0.5      MASS_7.3-65          irlba_2.3.5.1       
#> [49] pkgconfig_2.0.3      pillar_1.11.1        bslib_0.9.0         
#> [52] geigen_2.3           gtable_0.3.6         glue_1.8.0          
#> [55] Rcpp_1.1.1           xfun_0.55            tidyselect_1.2.1    
#> [58] svd_0.5.8            farver_2.1.2         htmltools_0.5.9     
#> [61] labeling_0.4.3       rmarkdown_2.30       compiler_4.5.1      
#> [64] S7_0.2.1