Generalized Orthogonal Procrustes Alignment

Performs Generalized Orthogonal Procrustes alignment to find orthogonal transformations. Aligns data from multiple domains by minimizing squared differences between corresponding task observations.

Usage

generalized_procrustes(data, ...)

# S3 method for class 'hyperdesign'
generalized_procrustes(
  data,
  y,
  preproc = center(),
  max_iter = 100,
  tol = 1e-06,
  tol_type = c("relative", "absolute"),
  verbose = FALSE,
  svd_method = c("irlba", "base"),
  svd_opts = list(),
  ...
)

Arguments

data

A hyperdesign object containing multiple data domains

...

Additional arguments (currently unused)

y

Name of the task/label variable to use for alignment

preproc

Preprocessing function to apply to the data (default: center())

max_iter

Maximum number of iterations (default: 100)

tol

Convergence tolerance (default: 1e-6)

tol_type

Type of tolerance check (default: "relative")

verbose

Whether to print progress messages (default: FALSE)

svd_method

SVD method to use (default: "irlba")

svd_opts

Options for SVD method (default: empty list)

Value

The return value depends on the specific method:

• For hyperdesign objects: A list containing orthogonal transformation matrices, consensus matrix, convergence information, and domain metadata

• For direct matrix input: A list with transformation matrices and alignment results

Details

This method extends classical Procrustes analysis to handle partial task observations, where each domain may observe only a subset of a global set of tasks. The algorithm uses an efficient Generalized Power Method (GPM) with sparse matrix operations and robust initialization to find optimal orthogonal transformations.

The Generalized Procrustes problem seeks to find orthogonal matrices \(O_i\) for each domain \(i\) that minimize: $$\sum_{i,j} \sum_{k \in T_{ij}} ||O_i^T A_i(:,k) - O_j^T A_j(:,k)||^2$$

where \(T_{ij}\) represents the set of tasks observed by both domains \(i\) and \(j\).

The algorithm handles several key challenges:

• Partial observations: Each domain may observe different subsets of tasks

• Sparse structure: Uses efficient sparse matrix operations for scalability

• Robust initialization: SVD-based initialization with fallback to random orthogonal matrices

• Convergence guarantees: Monotonic improvement with configurable tolerance

Key features:

• Vectorized operations using sparse matrix algebra

• Efficient projection onto the orthogonal group O(d)

• Optional tightness certificate for global optimality validation

• Flexible preprocessing and multiple input formats

References

Gower, J. C. (1975). Generalized procrustes analysis. Psychometrika, 40(1), 33-51.

Ten Berge, J. M. F. (1977). Orthogonal procrustes rotation for two or more matrices. Psychometrika, 42(2), 267-276.

See also

generalized_procrustes.hyperdesign

Examples

# \donttest{
# Example with hyperdesign data
library(multidesign)

# Create example domains with partial task overlap
d1_data <- matrix(rnorm(50), 5, 10)  # 5 tasks x 10 features
d1_design <- data.frame(task = factor(c("A", "B", "C", "D", "E")))
d1 <- multidesign(d1_data, d1_design)

d2_data <- matrix(rnorm(40), 4, 10)  # 4 tasks x 10 features
d2_design <- data.frame(task = factor(c("A", "C", "D", "F")))
d2 <- multidesign(d2_data, d2_design)

# Create hyperdesign
hd <- hyperdesign(list(domain1 = d1, domain2 = d2))

# Perform alignment
result <- generalized_procrustes(hd, task)
#> Warning: irlba::irlba failed: max(nu, nv) must be strictly less than min(nrow(A), ncol(A)). Falling back to random orthogonal matrices.

# Check convergence and results
print(result$converged)
#> [1] FALSE
print(dim(result$A_est))  # Features x total tasks
#> [1] 10  6
# }

# \donttest{
# Create example hyperdesign data
library(multidesign)

# Domain 1: 5 tasks x 10 features
d1_data <- matrix(rnorm(50), 5, 10)
d1_design <- data.frame(task = factor(c("A", "B", "C", "D", "E")))
d1 <- multidesign(d1_data, d1_design)

# Domain 2: 4 tasks x 10 features (partial overlap)
d2_data <- matrix(rnorm(40), 4, 10) 
d2_design <- data.frame(task = factor(c("A", "C", "D", "F")))
d2 <- multidesign(d2_data, d2_design)

# Create hyperdesign
hd <- hyperdesign(list(domain1 = d1, domain2 = d2))

# Perform alignment
result <- generalized_procrustes(hd, task)
#> Warning: irlba::irlba failed: max(nu, nv) must be strictly less than min(nrow(A), ncol(A)). Falling back to random orthogonal matrices.

# Access results
print(result$converged)
#> [1] FALSE
print(dim(result$A_est))  # 10 features x 6 total tasks
#> [1] 10  6
# }