Conditional Manifold Learning

Description

Find a low-dimensional embedding of high-dimensional data, conditioning on auxiliary manifold information. The current version supports conditional MDS and conditional ISOMAP.

Please cite this package as follows: Bui, A.T. (2024). Dimension Reduction with Prior Information for Knowledge Discovery. IEEE Transactions on Pattern Analysis and Machine Intelligence, 46, 3625-3636. https://doi.org/10.1109/TPAMI.2023.3346212

Details

Brief descriptions of the main functions of the package are provided below:

condMDS(): is the conditional MDS method, which uses conditional SMACOF to optimize its conditional stress function.

condIsomap(): is the conditional ISOMAP method, which is basically conditional MDS applying to graph distances (i.e., estimated geodesic distances) of the given distances/dissimilarities.

Author(s)

Anh Tuan Bui

Maintainer: Anh Tuan Bui <atbui@u.northwestern.edu>

References

Bui, A.T. (2024). Dimension Reduction with Prior Information for Knowledge Discovery. IEEE Transactions on Pattern Analysis and Machine Intelligence, 46, 3625-3636. https://doi.org/10.1109/TPAMI.2023.3346212

Bui, A. T. (2022). A Closed-Form Solution for Conditional Multidimensional Scaling. Pattern Recognition Letters 164, 148-152. https://doi.org/10.1016/j.patrec.2022.11.007

Bui, A.T. (2026). Conditional Multidimensional Scaling for Incomplete Conditioning Data. Journal of Multivariate Analysis. Accepted.

Examples

# Generate car-brand perception data
factor.weights <- c(90, 88, 83, 82, 81, 70, 68)/562
N <- 100
set.seed(1)
data <- matrix(runif(N*7), N, 7)
colnames(data) <- c('Quality', 'Safety', 'Value',	'Performance', 'Eco', 'Design', 'Tech')
rownames(data) <- paste('Brand', 1:N)
data.hat <- data + matrix(rnorm(N*7), N, 7)*data*.05
data.weighted <- t(apply(data, 1, function(x) x*factor.weights))
d <- dist(data.weighted)
d.hat <- d + rnorm(length(d))*d*.05

# Conditional MDS, using the first 4 factors as known features
u.cmds = condMDS(d.hat, data.hat[,1:4], 3, init='eigen')
u.cmds$B # compare with diag(factor.weights[1:4])
ccor(data.hat[,5:7], u.cmds$U)$cancor # canonical correlations
vegan::procrustes(data.hat[,5:7], u.cmds$U, symmetric = TRUE)$ss # Procrustes statistic

# Conditional ISOMAP, using the first 4 factors as known features
u.cisomap = condIsomap(d.hat, data.hat[,1:4], 3, k = 20, init='eigen')
u.cisomap$B # compare with diag(factor.weights[1:4])
ccor(data.hat[,5:7], u.cisomap$U)$cancor
vegan::procrustes(data.hat[,5:7], u.cisomap$U, symmetric = TRUE)$ss

# with missing values in V
V = data.hat[,1:4]
V[1, 1] = NA
u.cmds2 = condMDS(d.hat, V, 3, init='eigen')
u.cmds2$B # compare with diag(factor.weights[1:4])
ccor(data.hat[,5:7], u.cmds2$U)$cancor # canonical correlations
vegan::procrustes(data.hat[,5:7], u.cmds$U, symmetric = TRUE)$ss # Procrustes statistic
u.cmds2$V.hat[1,1] # imputed value for V[1, 1]; the ground truth is data[1,1]

Canonical Correlations

Description

Computes canonical correlations for two sets of multivariate data x and y.

Usage

ccor(x, y)

Arguments

Value

a list of the following components:

Author(s)

Anh Tuan Bui

Examples

ccor(iris[,1:2], iris[,3:4])

Conditional Euclidean distance

Description

Internal functions.

Usage

condDist(U, V.tilde, one_n_t=t(rep(1,nrow(U))))
condDist2(U, V.tilde2, one_n_t=t(rep(1,nrow(U))))

Arguments

Value

a dist object.

Author(s)

Anh Tuan Bui

References

Bui, A.T. (2024). Dimension Reduction with Prior Information for Knowledge Discovery. IEEE Transactions on Pattern Analysis and Machine Intelligence, 46, 3625-3636. https://doi.org/10.1109/TPAMI.2023.3346212

Conditional ISOMAP

Description

Finds a low-dimensional manifold embedding of a given distance/dissimilarity matrix, conditioning on auxiliary manifold parameters. The method applies conditional MDS (see condMDS) to a graph distance matrix computed for the given distances/dissimilarities, using the isomap{vegan} function.

Usage

condIsomap(d, V, u.dim, epsilon = NULL, k, W,
                        method = c('matrix', 'vector'), exact = TRUE,
                        it.max = 1000, gamma = 1e-05,
                        init = c('none', 'condSmacof', 'eigen', 'user'),
                        U.start, B.start,
                        V.tilde.start = NULL, ...)

Arguments

Value

Author(s)

Anh Tuan Bui

References

Bui, A.T. (2024). Dimension Reduction with Prior Information for Knowledge Discovery. IEEE Transactions on Pattern Analysis and Machine Intelligence, 46, 3625-3636. https://doi.org/10.1109/TPAMI.2023.3346212

Bui, A. T. (2022). A Closed-Form Solution for Conditional Multidimensional Scaling. Pattern Recognition Letters 164, 148-152. https://doi.org/10.1016/j.patrec.2022.11.007

Bui, A.T. (2026). Conditional Multidimensional Scaling for Incomplete Conditioning Data. Journal of Multivariate Analysis. Accepted.

See Also

condMDS

Examples

# see help(cml)

Conditional Multidimensional Scaling

Description

Wrapper of condSmacof, which finds a low-dimensional embedding of a given distance/dissimilarity matrix, conditioning on auxiliary manifold parameters.

Usage

condMDS(d, V, u.dim, W,
                     method = c('matrix', 'vector'), exact = TRUE,
                     it.max = 1000, gamma = 1e-05,
                     init = c('none', 'eigen', 'user'),
                     U.start, B.start, V.tilde.start = NULL) 

Arguments

Value

Author(s)

Anh Tuan Bui

References

Bui, A.T. (2024). Dimension Reduction with Prior Information for Knowledge Discovery. IEEE Transactions on Pattern Analysis and Machine Intelligence, 46, 3625-3636. https://doi.org/10.1109/TPAMI.2023.3346212

Bui, A. T. (2022). A Closed-Form Solution for Conditional Multidimensional Scaling. Pattern Recognition Letters 164, 148-152. https://doi.org/10.1016/j.patrec.2022.11.007

Bui, A.T. (2026). Conditional Multidimensional Scaling for Incomplete Conditioning Data. Journal of Multivariate Analysis. Accepted.

See Also

condSmacof, condSmacof_mer, condMDSeigen, condIsomap

Examples

# see help(cml)

Conditional Multidimensional Scaling With Closed-Form Solution

Description

Provides a closed-form solution for conditional multidimensional scaling, based on multiple linear regression and eigendecomposition.

Usage

condMDSeigen(d, V, u.dim, method = c('matrix', 'vector'))

Arguments

Value

Author(s)

Anh Tuan Bui

References

Bui, A. T. (2022). A Closed-Form Solution for Conditional Multidimensional Scaling. Pattern Recognition Letters 164, 148-152. https://doi.org/10.1016/j.patrec.2022.11.007

See Also

condMDS, condIsomap

Examples

# see help(cml)

Conditional SMACOF

Description

The conditional SMACOF algorithm. Intended for internal usage.

Usage

condSmacof(d, V, u.dim, W,
                        method = c('matrix', 'vector'), exact = TRUE,
                        it.max = 1000, gamma = 1e-05,
                        init = c('none', 'eigen', 'user'),
                        U.start, B.start)

Arguments

Value

Author(s)

Anh Tuan Bui

References

Bui, A.T. (2024). Dimension Reduction with Prior Information for Knowledge Discovery. IEEE Transactions on Pattern Analysis and Machine Intelligence, 46, 3625-3636. https://doi.org/10.1109/TPAMI.2023.3346212

Bui, A. T. (2022). A Closed-Form Solution for Conditional Multidimensional Scaling. Pattern Recognition Letters 164, 148-152. https://doi.org/10.1016/j.patrec.2022.11.007

Conditional SMACOF with incomplete conditioning data (missing entire row)

Description

Conditional SMACOF with missing known feature values. If a row has a missing value in a known feature, the algorithm treats all other known feature values of that row as missing. The algorithm also imputes the actual missing value(s). Intended for internal usage.

Usage

condSmacof_mer(d, V, u.dim, W,
           method = c('matrix', 'vector'), exact = TRUE,
           it.max = 1000, gamma = 1e-05,
           init = c('none', 'eigen', 'user'),
           U.start, B.start, V.tilde.start)

Arguments

Value

Author(s)

Anh Tuan Bui

References

Bui, A.T. (2024). Dimension Reduction with Prior Information for Knowledge Discovery. IEEE Transactions on Pattern Analysis and Machine Intelligence, 46, 3625-3636. https://doi.org/10.1109/TPAMI.2023.3346212

Bui, A. T. (2022). A Closed-Form Solution for Conditional Multidimensional Scaling. Pattern Recognition Letters 164, 148-152. https://doi.org/10.1016/j.patrec.2022.11.007

Bui, A.T. (2026). Conditional Multidimensional Scaling for Incomplete Conditioning Data. Journal of Multivariate Analysis. Accepted.

C(Z)

Description

Internal function.

Usage

cz(w, d, dz)

Arguments

Value

the matrix C(Z)

Author(s)

Anh Tuan Bui

References

Bui, A.T. (2024). Dimension Reduction with Prior Information for Knowledge Discovery. IEEE Transactions on Pattern Analysis and Machine Intelligence, 46, 3625-3636. https://doi.org/10.1109/TPAMI.2023.3346212

Moore-Penrose Inverse

Description

Computes the Moore-Penrose inverse (a.k.a., generalized inverse or pseudoinverse) of a matrix based on singular-value decomposition (SVD).

Usage

mpinv(A, eps = sqrt(.Machine$double.eps))

Arguments

Value

the Moore-Penrose inverse.

Author(s)

Anh Tuan Bui

Examples

mpinv(2*diag(4))