| Title: | Canonical Correlation Analysis via Reduced Rank Regression |
| Version: | 0.1.0 |
| Date: | 2025-08-22 |
| Author: | Claire Donnat 
[aut, cre],
Elena Tuzhilina
[aut],
Zixuan Wu [aut] |
| Maintainer: | Claire Donnat <cdonnat@uchicago.edu> |
| Description: | Canonical correlation analysis (CCA) via reduced-rank regression with support for regularization and cross-validation. Several methods for estimating CCA in high-dimensional settings are implemented. The first set of methods, cca_rrr() (and variants: cca_group_rrr() and cca_graph_rrr()), assumes that one dataset is high-dimensional and the other is low-dimensional, while the second, ecca() (for Efficient CCA) assumes that both datasets are high-dimensional. For both methods, standard l1 regularization as well as group-lasso regularization are available. cca_graph_rrr further supports total variation regularization when there is a known graph structure among the variables of the high-dimensional dataset. In this case, the loadings of the canonical directions of the high-dimensional dataset are assumed to be smooth on the graph. For more details see Donnat and Tuzhilina (2024) <doi:10.48550/arXiv.2405.19539> and Wu, Tuzhilina and Donnat (2025) <doi:10.48550/arXiv.2507.11160>. |
| Depends: | R (≥ 3.5.0) |
| Imports: | purrr, magrittr, tidyr, dplyr, foreach, pracma, corpcor, matrixStats, RSpectra, caret |
| Suggests: | SMUT, igraph, testthat (≥ 3.0.0), rrpack, CVXR, Matrix, glmnet, CCA, PMA, doParallel, crayon |
| License: | MIT + file LICENSE |
| Encoding: | UTF-8 |
| RoxygenNote: | 7.3.2 |
| Config/testthat/edition: | 3 |
| NeedsCompilation: | no |
| Packaged: | 2025-09-11 21:48:50 UTC; clairedonnat |
| Repository: | CRAN |
| Date/Publication: | 2025-09-16 08:00:07 UTC |
False Positive Rate (TPR)
Description
This is a function that compares the structure of two matrices A and B. It outputs the number of entries where A is not zero but Bis. A and B need to have the same number of rows and columns
Usage
FPR(A, B, tol = 1e-04)
Arguments
A |
A matrix. |
B |
A matrix (assumed to be the ground truth). |
tol |
tolerance for declaring the entries non zero. |
Value
False Positive Rate (nb of values that are non zero in A and zero in B / (nb of values that are non zero in A))
Examples
A <- matrix(c(1, 0, 0, 1, 1, 0), nrow = 2)
B <- matrix(c(1, 0, 1, 1, 0, 0), nrow = 2)
FPR(A, B)
Function to perform Sparse CCA based on Waaijenborg et al. (2008) REFERENCE Parkhomenko et al. (2009), "Sparse Canonical Correlation Anlaysis with Application to Genomic Data Integration" in Statistical Applications in Genetics and Molecular Biology, Volume 8, Issue 1, Article 1
Description
Function to perform Sparse CCA based on Waaijenborg et al. (2008) REFERENCE Parkhomenko et al. (2009), "Sparse Canonical Correlation Anlaysis with Application to Genomic Data Integration" in Statistical Applications in Genetics and Molecular Biology, Volume 8, Issue 1, Article 1
Usage
SCCA_Parkhomenko(
x.data,
y.data,
n.cv = 5,
lambda.v.seq = seq(0, 0.2, by = 0.02),
lambda.u.seq = seq(0, 0.2, by = 0.02),
Krank = 1,
standardize = TRUE
)
Arguments
x.data |
Matrix of predictors (n x p) |
y.data |
Matrix of responses (n x q) |
n.cv |
Number of cross-validation folds (default is 5) |
lambda.v.seq |
Vector of sparsity parameters for Y (default is a sequence from 0 to 1 with step 0.1) |
lambda.u.seq |
Vector of sparsity parameters for X (default is a sequence from 0 to 1 with step 0.1) |
Krank |
Number of canonical components to extract |
standardize |
Standardize (center and scale) the data matrices X and Y (default is TRUE) before analysis |
Value
A list with elements:
- U
Canonical direction matrix for X (p x r)
- V
Canonical direction matrix for Y (q x r)
- cor
Canonical correlations
Function to perform Sparse CCA based on Wilms and Croux (2018) REFERENCE Wilms, I., & Croux, C. (2018). Sparse canonical correlation analysis using alternating regressions. Journal of Computational and Graphical Statistics, 27(1), 1-10.
Description
Function to perform Sparse CCA based on Wilms and Croux (2018) REFERENCE Wilms, I., & Croux, C. (2018). Sparse canonical correlation analysis using alternating regressions. Journal of Computational and Graphical Statistics, 27(1), 1-10.
Usage
SparseCCA(
X,
Y,
lambdaAseq = seq(from = 1, to = 0.01, by = -0.01),
lambdaBseq = seq(from = 1, to = 0.01, by = -0.01),
rank,
selection.criterion = 1,
n.cv = 5,
A.initial = NULL,
B.initial = NULL,
max.iter = 20,
conv = 10^-2,
standardize = TRUE
)
Arguments
X |
Matrix of predictors (n x p) |
Y |
Matrix of responses (n x q) |
lambdaAseq |
Vector of sparsity parameters for X (default is a sequence from 0 to 1 with step 0.1) |
lambdaBseq |
Vector of sparsity parameters for Y (default is a sequence from 0 to 1 with step 0.1) |
rank |
Number of canonical components to extract |
selection.criterion |
Criterion for selecting the optimal tuning parameter (1 for minimizing difference between test and training sample correlation, 2 for maximizing test sample correlation) |
n.cv |
Number of cross-validation folds (default is 5) |
A.initial |
Initial value for the canonical vector A (default is NULL, which uses a canonical ridge solution) |
B.initial |
Initial value for the canonical vector B (default is NULL, which uses a canonical ridge solution) |
max.iter |
Maximum number of iterations for convergence (default is 20) |
conv |
Convergence threshold (default is 1e-2) |
standardize |
Standardize (center and scale) the data matrices X and Y (default is TRUE) before analysis |
Value
A list with elements:
- U
Canonical direction matrix for X (p x r)
- V
Canonical direction matrix for Y (q x r)
- loss
Mean squared error of prediction
- cor
Canonical covariances
True Negative Rate (TNR)
Description
This is a function that compares the structure of two matrices A and B. It outputs the number of entries where A and B are both 0. A and B need to have the same number of rows and columns
Usage
TNR(A, B, tol = 1e-04)
Arguments
A |
A matrix. |
B |
A matrix (assumed to be the ground truth).. |
tol |
tolerance for declaring the entries non zero. |
Value
True Negative Rate (nb of values that are zero in A and zero in B / (nb of values that are zero in A))
True Positive Rate (TPR)
Description
This is a function that compares the structure of two matrices A and B. It outputs the number of entries that A and B have in common that are different from zero. A and B need to have the same number of rows and columns
Usage
TPR(A, B, tol = 1e-04)
Arguments
A |
A matrix (the estimator). |
B |
A matrix (assumed to be the ground truth). |
tol |
tolerance for declaring the entries non zero. |
Value
True Positive Rate (nb of values that are non zero in both A and B / (nb of values that are non zero in A))
Examples
A <- matrix(c(1, 0, 0, 1, 1, 0), nrow = 2)
B <- matrix(c(1, 0, 1, 1, 0, 0), nrow = 2)
TPR(A, B)
Sparse CCA by Witten and Tibshirani (2009)
Description
Sparse CCA by Witten and Tibshirani (2009)
Usage
Witten.CV(
X,
Y,
n.cv = 5,
rank,
lambdax = matrix(seq(from = 0, to = 1, by = 0.1), nrow = 1),
lambday = matrix(seq(from = 0, to = 1, by = 0.1), nrow = 1),
standardize = TRUE
)
Arguments
X |
Matrix of predictors (n x p) |
Y |
Matrix of responses (n x q) |
n.cv |
Number of cross-validation folds (default is 5) |
rank |
Number of canonical components to extract |
lambdax |
Vector of sparsity parameters for X (default is a sequence from 0 to 1 with step 0.1) |
lambday |
Vector of sparsity parameters for Y (default is a sequence from 0 to 1 with step 0.1) |
standardize |
Standardize (center and scale) the data matrices X and Y (default is TRUE) before analysis |
Value
the appropriate levels of regularisation
Graph-regularized Reduced-Rank Regression for Canonical Correlation Analysis
Description
Solves a sparse canonical correlation problem using a graph-constrained reduced-rank regression formulation. The problem is solved via an ADMM approach.
Usage
cca_graph_rrr(
X,
Y,
Gamma,
Sx = NULL,
Sy = NULL,
Sxy = NULL,
lambda = 0,
r,
standardize = FALSE,
LW_Sy = TRUE,
rho = 10,
niter = 10000,
thresh = 1e-04,
thresh_0 = 1e-06,
verbose = FALSE,
Gamma_dagger = NULL
)
Arguments
X |
Matrix of predictors (n x p) |
Y |
Matrix of responses (n x q) |
Gamma |
Graph constraint matrix (g x p) |
Sx |
Optional covariance matrix for X. If NULL, computed as t(X) %*% X / n |
Sy |
Optional covariance matrix for Y. If NULL, computed similarly; optionally shrunk via Ledoit-Wolf |
Sxy |
Optional cross-covariance matrix (not currently used) |
lambda |
Regularization parameter for sparsity |
r |
Target rank |
standardize |
Whether to center and scale X and Y (default FALSE = center only) |
LW_Sy |
Whether to apply Ledoit-Wolf shrinkage to Sy |
rho |
ADMM penalty parameter |
niter |
Maximum number of ADMM iterations |
thresh |
Convergence threshold for ADMM |
thresh_0 |
Threshold for small values in the coefficient matrix (default 1e-6) |
verbose |
Whether to print diagnostic output |
Gamma_dagger |
Optional pseudoinverse of Gamma (computed if NULL) |
Value
A list with elements:
- U
Canonical direction matrix for X (p x r)
- V
Canonical direction matrix for Y (q x r)
- cor
Canonical covariances
- loss
The prediction error 1/n * \| XU - YV\|^2
Graph-regularized Reduced-Rank Regression for Canonical Correlation Analysis with cross validation
Description
Solves a sparse canonical correlation problem using a graph-constrained reduced-rank regression formulation. The problem is solved via an ADMM approach.
Usage
cca_graph_rrr_cv(
X,
Y,
Gamma,
r = 2,
lambdas = 10^seq(-3, 1.5, length.out = 10),
kfolds = 5,
parallelize = FALSE,
standardize = TRUE,
LW_Sy = FALSE,
rho = 10,
niter = 10000,
thresh = 1e-04,
thresh_0 = 1e-06,
verbose = FALSE,
Gamma_dagger = NULL,
nb_cores = NULL
)
Arguments
X |
Matrix of predictors (n x p) |
Y |
Matrix of responses (n x q) |
Gamma |
Graph constraint matrix (g x p) |
r |
Target rank |
lambdas |
Grid of regularization parameters to test for sparsity |
kfolds |
Number of folds for cross-validation |
parallelize |
Whether to parallelize cross-validation |
standardize |
Whether to center and scale X and Y (default FALSE = center only) |
LW_Sy |
Whether to apply Ledoit-Wolf shrinkage to Sy |
rho |
ADMM penalty parameter |
niter |
Maximum number of ADMM iterations |
thresh |
Convergence threshold for ADMM |
thresh_0 |
Threshold for small values in the coefficient matrix (default 1e-6) |
verbose |
Whether to print diagnostic output |
Gamma_dagger |
Optional pseudoinverse of Gamma (computed if NULL) |
nb_cores |
Number of cores to use for parallelization (default is all available cores minus 1) |
Value
A list with elements:
- U
Canonical direction matrix for X (p x r)
- V
Canonical direction matrix for Y (q x r)
- lambda
Optimal regularisation parameter lambda chosen by CV
- rmse
Mean squared error of prediction (as computed in the CV)
- cor
Canonical covariances
Group-Sparse Canonical Correlation via Reduced-Rank Regression
Description
Performs group-sparse reduced-rank regression for CCA using either ADMM or CVXR solvers.
Usage
cca_group_rrr(
X,
Y,
groups,
Sx = NULL,
Sy = NULL,
Sxy = NULL,
lambda = 0,
r,
standardize = FALSE,
LW_Sy = TRUE,
solver = "ADMM",
rho = 1,
niter = 10000,
thresh = 1e-04,
thresh_0 = 1e-06,
verbose = FALSE
)
Arguments
X |
Predictor matrix (n x p) |
Y |
Response matrix (n x q) |
groups |
List of index vectors defining groups of predictors |
Sx |
Optional covariance matrix for X; if NULL computed internally |
Sy |
Optional covariance matrix for Y; if NULL computed internally |
Sxy |
Optional cross covariance matrix for X and Y; if NULL computed internally |
lambda |
Regularization parameter |
r |
Target rank |
standardize |
Whether to scale variables |
LW_Sy |
Whether to apply Ledoit-Wolf shrinkage to Sy (default TRUE) |
solver |
Either "ADMM" or "CVXR" |
rho |
ADMM parameter |
niter |
Maximum number of ADMM iterations |
thresh |
Convergence threshold for ADMM |
thresh_0 |
tolerance for declaring entries non-zero |
verbose |
Print diagnostics |
Value
A list with elements:
- U
Canonical direction matrix for X (p x r)
- V
Canonical direction matrix for Y (q x r)
- cor
Canonical covariances
- loss
The prediction error 1/n * \| XU - YV\|^2
Group-Sparse Canonical Correlation via Reduced-Rank Regression with CV
Description
Performs group-sparse reduced-rank regression for CCA using either ADMM or CVXR solvers.
Usage
cca_group_rrr_cv(
X,
Y,
groups,
r = 2,
lambdas = 10^seq(-3, 1.5, length.out = 10),
kfolds = 5,
parallelize = FALSE,
standardize = FALSE,
LW_Sy = TRUE,
solver = "ADMM",
rho = 1,
thresh_0 = 1e-06,
niter = 10000,
thresh = 1e-04,
verbose = FALSE,
nb_cores = NULL
)
Arguments
X |
Predictor matrix (n x p) |
Y |
Response matrix (n x q) |
groups |
List of index vectors defining groups of predictors |
r |
Target rank |
lambdas |
Grid of regularization parameters to try out |
kfolds |
Nb of folds for the CV procedure |
parallelize |
Whether to use parallel processing (default is FALSE) |
standardize |
Whether to scale variables |
LW_Sy |
Whether to apply Ledoit-Wolf shrinkage to Sy (default TRUE) |
solver |
Either "ADMM" or "CVXR" |
rho |
ADMM parameter |
thresh_0 |
tolerance for declaring entries non-zero |
niter |
Maximum number of ADMM iterations |
thresh |
Convergence threshold for ADMM |
verbose |
Print diagnostics |
nb_cores |
Number of cores to use for parallelization (default is all available cores minus 1) |
Value
A list with elements:
- U
Canonical direction matrix for X (p x r)
- V
Canonical direction matrix for Y (q x r)
- lambda
Optimal regularisation parameter lambda chosen by CV
- rmse
Mean squared error of prediction (as computed in the CV)
- cor
Canonical covariances
Canonical Correlation Analysis via Reduced Rank Regression (RRR)
Description
Estimates canonical directions using various RRR solvers and penalties.
Usage
cca_rrr(
X,
Y,
Sx = NULL,
Sy = NULL,
lambda = 0,
r,
highdim = TRUE,
solver = "ADMM",
LW_Sy = TRUE,
standardize = TRUE,
rho = 1,
niter = 10000,
thresh = 1e-04,
thresh_0 = 1e-06,
verbose = FALSE
)
Arguments
X |
Matrix of predictors. |
Y |
Matrix of responses. |
Sx |
Optional X covariance matrix. |
Sy |
Optional Y covariance matrix. |
lambda |
Regularization parameter. |
r |
Rank of the solution. |
highdim |
Boolean for high-dimensional regime. |
solver |
Solver type: "rrr", "CVX", or "ADMM". |
LW_Sy |
Whether to use Ledoit-Wolf shrinkage for Sy. |
standardize |
Logical; should X and Y be scaled. |
rho |
ADMM parameter. |
niter |
Maximum number of iterations for ADMM. |
thresh |
Convergence threshold. |
thresh_0 |
For the ADMM solver: Set entries whose absolute value is below this to 0 (default 1e-6). |
verbose |
Logical for verbose output. |
Value
A list with elements:
U: Canonical direction matrix for X (p x r)
V: Canonical direction matrix for Y (q x r)
cor: Canonical covariances
loss: The prediction error 1/n * || XU - YV ||^2
Cross-validated Canonical Correlation Analysis via RRR
Description
Performs cross-validation to select optimal lambda, fits CCA_rrr. Canonical Correlation Analysis via Reduced Rank Regression (RRR)
Usage
cca_rrr_cv(
X,
Y,
r = 2,
lambdas = 10^seq(-3, 1.5, length.out = 100),
kfolds = 14,
solver = "ADMM",
parallelize = FALSE,
LW_Sy = TRUE,
standardize = TRUE,
rho = 1,
thresh_0 = 1e-06,
niter = 10000,
thresh = 1e-04,
verbose = FALSE,
nb_cores = NULL
)
Arguments
X |
Matrix of predictors. |
Y |
Matrix of responses. |
r |
Rank of the solution. |
lambdas |
Sequence of lambda values for cross-validation. |
kfolds |
Number of folds for cross-validation. |
solver |
Solver type: "rrr", "CVX", or "ADMM". |
parallelize |
Logical; should cross-validation be parallelized? |
LW_Sy |
Whether to use Ledoit-Wolf shrinkage for Sy. |
standardize |
Logical; should X and Y be scaled. |
rho |
ADMM parameter. |
thresh_0 |
tolerance for declaring entries non-zero |
niter |
Maximum number of iterations for ADMM. |
thresh |
Convergence threshold. |
verbose |
Logical for verbose output. |
nb_cores |
Number of cores to use for parallelization (default is all available cores minus 1) |
Value
A list with elements:
U: Canonical direction matrix for X (p x r)
V: Canonical direction matrix for Y (q x r)
lambda: Optimal regularisation parameter lambda chosen by CV
rmse: Mean squared error of prediction (as computed in the CV)
cor: Canonical correlations
Sparse Canonical Correlation via Reduced-Rank Regression when both X and Y are high-dimensional.
Description
Performs group-sparse reduced-rank regression for CCA using either ADMM or CVXR solvers.
Usage
ecca(
X,
Y,
lambda = 0,
groups = NULL,
Sx = NULL,
Sy = NULL,
Sxy = NULL,
r = 2,
standardize = FALSE,
rho = 1,
B0 = NULL,
eps = 1e-04,
maxiter = 500,
verbose = TRUE
)
Arguments
X |
Predictor matrix (n x p) |
Y |
Response matrix (n x q) |
lambda |
Regularization parameter |
groups |
List of index vectors defining groups of predictors |
Sx |
precomputed covariance matrix for X (optional) |
Sy |
precomputed covariance matrix for Y (optional) |
Sxy |
precomputed covariance matrix between X and Y (optional) |
r |
Target rank |
standardize |
Whether to scale variables |
rho |
ADMM parameter |
B0 |
Initial value for the coefficient matrix (optional) |
eps |
Convergence threshold for ADMM |
maxiter |
Maximum number of ADMM iterations |
verbose |
Print diagnostics |
Value
A list with elements:
- U
Canonical direction matrix for X (p x r)
- V
Canonical direction matrix for Y (q x r)
- cor
Canonical covariances
- loss
The prediction error 1/n * \| XU - YV\|^2
Sparse Canonical Correlation via Reduced-Rank Regression when both X and Y are high-dimensional, with Cross-Validation
Description
Performs group-sparse reduced-rank regression for CCA using either ADMM or CVXR solvers.
Usage
ecca.cv(
X,
Y,
lambdas = 0,
groups = NULL,
r = 2,
standardize = FALSE,
rho = 1,
B0 = NULL,
nfold = 5,
select = "lambda.min",
eps = 1e-04,
maxiter = 500,
verbose = FALSE,
parallel = FALSE,
nb_cores = NULL,
set_seed_cv = NULL,
scoring_method = "mse",
cv_use_median = FALSE
)
Arguments
X |
Predictor matrix (n x p) |
Y |
Response matrix (n x q) |
lambdas |
Choice of regularization parameter |
groups |
List of index vectors defining groups of predictors |
r |
Target rank |
standardize |
Whether to scale variables |
rho |
ADMM parameter |
B0 |
Initial value for the coefficient matrix (optional) |
nfold |
Number of cross-validation folds |
select |
Which lambda to select: "lambda.min" or "lambda.1se" |
eps |
Convergence threshold for ADMM |
maxiter |
Maximum number of ADMM iterations |
verbose |
Print diagnostics |
parallel |
Whether to run cross-validation in parallel |
nb_cores |
Number of cores to use for parallel processing (default is NULL, which uses all available cores) |
set_seed_cv |
Optional seed for reproducibility of cross-validation folds (de) |
scoring_method |
Method to score the model during cross-validation, either "mse" (mean squared error) or "trace" (trace of the product of matrices) |
cv_use_median |
Whether to use the median of the cross-validation scores instead of the mean. Default is FALSE. |
Value
A list with elements:
- U
Canonical direction matrix for X (p x r)
- V
Canonical direction matrix for Y (q x r)
- cor
Canonical covariances
- loss
The prediction error 1/n * \| XU - YV\|^2
Return the edge incidence matrix of an igraph graph
Description
Return the edge incidence matrix of an igraph graph
Usage
get_edge_incidence(g, weight = 1)
Arguments
g |
igraph graph object. |
weight |
edge weights. |
Value
Edge incidence matrix of the graph g, with +weight for the source node and -weight for the target node.
Metrics for subspaces
Description
Calculate principal angles between subspace spanned by the columns of a and the subspace spanned by the columns of b
Usage
principal_angles(a, b)
Arguments
a |
A matrix whose columns span a subspace. |
b |
A matrix whose columns span a subspace. |
Value
a vector of principal angles (in radians)
Examples
a <- matrix(rnorm(9), 3, 3)
b <- matrix(rnorm(9), 3, 3)
principal_angles(a, b)
Function to perform regular (low dimensional) canonical correlation analysis (CCA
Description
Function to perform regular (low dimensional) canonical correlation analysis (CCA
Usage
regular_cca(X, Y, rank)
Arguments
X |
Matrix of predictors (n x p) |
Y |
Matrix of responses (n x q) |
rank |
Number of canonical components to extract |
Value
A list with elements:
- U
Canonical direction matrix for X (p x r)
- V
Canonical direction matrix for Y (q x r)
- cor
Canonical covariances
Set up a parallel backend with graceful fallbacks.
Description
Attempts to create a parallel cluster, first trying the efficient FORK method (on Unix-like systems), then falling back to PSOCK, and finally returning NULL if all attempts fail.
Usage
setup_parallel_backend(num_cores = NULL, verbose = FALSE)
Arguments
num_cores |
The number of cores to use. If NULL, it's determined automatically. |
verbose |
If TRUE, prints messages about the setup process. |
Value
A cluster object cl on success, or NULL on failure.
SinTheta distance between subspaces
Description
Calculate the distance spanned by the columns of A and the subspace spanned by the columns of B, defined as ||UU^T - VV^T||_F / sqrt(2)
Usage
sinTheta(U, V)
Arguments
U |
A matrix whose columns span a subspace. |
V |
A matrix whose columns span a subspace. |
Value
sinTheta distance between the two subspaces spanned by the matrices A and B, defined as ||UU^T - VV^T||_F / sqrt(2)
Additional Benchmarks for Sparse CCA Methods
Description
Additional Benchmarks for Sparse CCA Methods
Usage
sparse_CCA_benchmarks(
X_train,
Y_train,
S = NULL,
rank = 2,
kfolds = 5,
method.type = "FIT_SAR_CV",
lambdax = 10^seq(from = -3, to = 2, length = 10),
lambday = c(0, 1e-07, 1e-06, 1e-05),
standardize = TRUE
)
Arguments
X_train |
Matrix of predictors (n x p) |
Y_train |
Matrix of responses (n x q) |
S |
Optional covariance matrix (default is NULL, which computes it from X_train and Y_train) |
rank |
Target rank for the CCA (default is 2) |
kfolds |
Number of cross-validation folds (default is 5) |
method.type |
Type of method to use for Sparse CCA (default is "FIT_SAR_CV"). Choices include "FIT_SAR_BIC", "FIT_SAR_CV", "Witten_Perm", "Witten.CV", and "SCCA_Parkhomenko". |
lambdax |
Vector of sparsity parameters for X (default is a sequence from 0 to 1 with step 0.1) |
lambday |
Vector of sparsity parameters for Y (default is a sequence from 0 to 1 with step 0.1) |
standardize |
Standardize (center and scale) the data matrices X and Y (default is TRUE) before analysis |
Value
A matrix (p+q)x r containing the canonical directions for X and Y.
Subdistance between subspaces
Description
Calculate subdistance between subspace spanned by the columns of a and the subspace spanned by the columns of b
Usage
subdistance(A, B)
Arguments
A |
A matrix whose columns span a subspace. |
B |
A matrix whose columns span a subspace. |
Value
subdistance between the two subspaces spanned by the matrices A and B, defined as min(O orthogonal) ||AO-B||_F