Coupled Diagonalization for Multi-Modal Alignment

Performs Coupled Diagonalization alignment on multi-modal data. Finds coupled eigenvectors across multiple modalities that jointly diagonalize graph Laplacians while respecting cross-modal correspondences.

Performs Coupled Diagonalization alignment on hyperdesign data structures containing multiple modalities. Finds coupled eigenvectors that jointly diagonalize graph Laplacians while respecting cross-modal correspondences.

Usage

coupled_diagonalization(data, ...)

# S3 method for class 'hyperdesign'
coupled_diagonalization(
  data,
  correspondence = NULL,
  preproc = center(),
  ncomp = 10,
  ncomp_per_domain = 20,
  mu_coupling = 1,
  alpha_match = 0,
  lambda_target = NULL,
  knn = 10,
  sigma = NULL,
  spectral_cache = NULL,
  max_iter = 200,
  step_size = 0.3,
  tol = 1e-06,
  verbose = FALSE,
  ...
)

# Default S3 method
coupled_diagonalization(data, ...)

Arguments

data: A hyperdesign object containing multiple data domains
...: Additional arguments (currently unused)
correspondence: Optional list of correspondence matrices between modalities. If NULL (default), assumes full correspondence based on sample order. Each matrix should be n_i x n_j sparse matrix indicating correspondences between modality i and j.
preproc: Preprocessing function to apply to the data (default: center())
ncomp: Number of coupled eigenvectors to extract (default: 10)
ncomp_per_domain: Number of eigenvectors per domain to compute before coupling (default: 20). Must be >= ncomp.
mu_coupling: Coupling weight controlling alignment vs diagonalization trade-off (default: 1). Higher values emphasize cross-modal alignment.
alpha_match: Optional weight for matching the diagonal of A_i^T Lambda_i A_i to target eigenvalues (default: 0).
lambda_target: Optional list of length m providing target eigenvalue vectors for each modality (only the first ncomp entries are used).
knn: Number of nearest neighbors for graph construction (default: 10)
sigma: Gaussian kernel bandwidth for graph weights. If NULL (default), uses median of nearest neighbor distances.
spectral_cache: Optional precomputed spectral cache to reuse Laplacian subspaces across repeated runs (for example previous_fit$diagnostics$spectral_cache). When supplied, graph/Laplacian/eigendecomposition steps are skipped.
max_iter: Maximum iterations for Stiefel manifold optimization (default: 200)
step_size: Base step size for gradient descent (default: 0.3)
tol: Convergence tolerance for relative cost change (default: 1e-6)
verbose: Whether to print convergence information (default: FALSE)

Value

A multiblock_biprojector object containing:

s: Coupled eigenvectors (scores) for all samples
v: Projection matrices for out-of-sample extension
coupled_bases: List of coupled bases for each modality
sdev: Standard deviations of the coupled components
Additional metadata for reconstruction and diagnostics

A multiblock_biprojector object containing:

s: Coupled eigenvectors (scores) concatenated across domains
v: Projection matrices for out-of-sample extension
coupled_bases: List of coupled bases V_i for each modality
sdev: Standard deviations of coupled components
eigenvalues: List of eigenvalues for each domain
diagnostics: List with per-iteration cost diagnostics
converged: Logical indicating convergence
iterations: Number of iterations performed
final_cost: Final objective value

Details

Coupled Diagonalization (Eynard et al. 2015) extends spectral methods to multi-modal data by finding bases that simultaneously diagonalize individual Laplacians while being aligned through correspondence constraints. The algorithm uses Stiefel manifold optimization to maintain orthogonality constraints.

The algorithm minimizes the objective function: $$F(A) = sum_i ||A_i^T L_i A_i - diag(A_i^T L_i A_i)||_F^2 + mu_c sum_{i<j} ||F_i^T U_i A_i - F_j^T U_j A_j||_F^2$$

where:

A_i are the coupling matrices on the Stiefel manifold
L_i are the eigenvalues of the i-th Laplacian
U_i are the eigenvectors of the i-th Laplacian
F_i are the correspondence matrices between modalities
mu_c controls the coupling strength

The first term encourages diagonalization of individual Laplacians, while the second term enforces alignment through correspondences. The algorithm uses projected gradient descent on the Stiefel manifold to maintain orthogonality.

Key features:

Handles partial correspondences between modalities
Preserves manifold structure through graph Laplacians
Maintains orthogonality via Stiefel manifold optimization
Supports semi-supervised learning with missing correspondences

The algorithm minimizes the objective


F = sum_i ||A_i^T Lambda_i A_i - diag(A_i^T Lambda_i A_i)||_F^2 +
    mu_c sum_{i<j} ||F_i^T U_i A_i - F_j^T U_j A_j||_F^2

It uses projected gradient descent on the Stiefel manifold to maintain orthogonality of the coupling matrices A_i. When alpha_match > 0, the optional match term nudges the diagonal entries of A_i^T Lambda_i A_i toward user-specified target eigenvalues (without adding extra off-diagonal penalty).

References

Eynard, D., Kovnatsky, A., Bronstein, M. M., Glashoff, K., & Bronstein, A. M. (2015). Multimodal manifold analysis by simultaneous diagonalization of Laplacians. IEEE Transactions on Pattern Analysis and Machine Intelligence, 37(12), 2505-2517.

Eynard et al. (2015). Multimodal manifold analysis by simultaneous diagonalization of Laplacians. IEEE TPAMI, 37(12), 2505-2517.

Examples