landmark_mds {bigmds}

Landmark MDS

Description

Landmark MDS (LMDS) algorithm applies first classical MDS to a subset of the data (landmark points) and then the remaining individuals are projected onto the landmark low dimensional configuration using a distance-based triangulation procedure.

Usage

landmark_mds(x, num_landmarks, r)

Arguments

x

A matrix with n points (rows) and k variables (columns).

num_landmarks

Number of landmark points to obtain an initial MDS configuration. It is equivalent to l parameter used in interpolation_mds(), divide_conquer_mds() and fast_mds(). Therefore, it is the size for which classical MDS can be computed efficiently (using cmdscale function). It means that if \bar{l} is the limit size for which classical MDS is applicable, then l\leq \bar{l}.

r

Number of principal coordinates to be extracted.

Details

LMDS applies first classical MDS to a subset of the data (landmark points). Then, it uses a distance-based triangulation procedure to project the non-landmark individuals. This distance-based triangulation procedure coincides with Gower's interpolation formula.

This method is similar to interpolation_mds() and reduced_mds().

Value

Returns a list containing the following elements:

points

A matrix that consists of n points (rows) and r variables (columns) corresponding to the principal coordinates. Since a dimensionality reduction is performed, r<<k

eigen

The first r largest eigenvalues: \lambda_i, i \in \{1, \dots, r\} , where each \lambda_i is obtained from applying classical MDS to the first data subset.

References

Delicado P. and C. Pachón-García (2021). Multidimensional Scaling for Big Data. https://arxiv.org/abs/2007.11919.

Borg, I. and P. Groenen (2005). Modern Multidimensional Scaling: Theory and Applications. Springer.

De Silva V. and JB. Tenenbaum (2004). Sparse multidimensional scaling using landmark points. Technical Report, Stanford University.

Gower JC. (1968). Adding a point to vector diagrams in multivariate analysis. Biometrika.

Examples

set.seed(42)
x <- matrix(data = rnorm(4 * 10000), nrow = 10000) %*% diag(c(9, 4, 1, 1))
mds <- landmark_mds(x = x, num_landmarks = 200, r = 2)
head(mds$points)
mds$eigen