Introduction
In this vignette, we consider a novel graph embedding method, Vicus
(Wang B 2017).
Here, we use the Swiss roll data, which is a well known toy
model.
set.seed(1)
N <- 300
p <- sqrt(2 + 2 * seq(-1, 1 - 2 / N, 2 / N))
y <- 2 * runif(N, -1, 1)
X <- cbind(p * cos(2 * pi * p), y, p * sin(2 * pi * p))
X <- scale(X, center=TRUE, scale=TRUE) * 3
labelX <- c(rep(1:11, each = floor(N / 11)), rep(11, length=3))
library("scatterplot3d")
# Color Setting
colors <- labelX
cols <- c("#9E0142", "#D53E4F", "#F46D43", "#FDAE61",
"#FEE08B", "#FFFFBF", "#E6F598", "#ABDDA4",
"#66C2A5", "#3288BD", "#5E4FA2")
for(i in seq_along(cols)){
colors[which(colors == i)] <- cols[i]
}
oldpar <- par("cex")
par(cex = 1.2)
scatterplot3d(X, color=colors, pch=16, main="Original Data", angle=40)
2D Embedding
The Vicus package provides three types of graph
embedding algorithms: Vicus, Laplacian Eigenmaps
(LEM), and Hessian Locally Linear Embedding
(HLLE).
First, the graphMatrix function computes a matrix
containing graph information for each algorithm:
library("Vicus")
objVicus <- graphMatrix(X, algorithm="Vicus", ndim=2, K=10)
objLEM <- graphMatrix(X, algorithm="LEM", ndim=2, K=10)
objHLLE <- graphMatrix(X, algorithm="HLLE", ndim=2, K=5)
str(objVicus, 2)
## List of 3
## $ M :Formal class 'dgCMatrix' [package "Matrix"] with 6 slots
## $ algorithm: chr "Vicus"
## $ ndim : num 2
## List of 3
## $ M :Formal class 'dgCMatrix' [package "Matrix"] with 6 slots
## $ algorithm: chr "LEM"
## $ ndim : num 2
## List of 3
## $ M :Formal class 'dgCMatrix' [package "Matrix"] with 6 slots
## $ algorithm: chr "HLLE"
## $ ndim : num 2
Next, the embedding function performs eigenvalue
decomposition and estimates the low-dimensional coordinates.
outVicus <- embedding(objVicus)
outLEM <- embedding(objLEM)
outHLLE <- embedding(objHLLE)
The low dimensional coordinates show that Vicus is better able to
capture the local structure of the Swiss roll data.
layout(t(1:3))
plot(outVicus, col=colors, pch=16, main="Vicus", cex=2)
plot(outLEM, col=colors, pch=16, main="LEM", cex=2)
plot(outHLLE, col=colors, pch=16, main="HLLE", cex=2)
3D Embedding
It can also be embedded to any dimension by simply changing the value
of ndim as follows:
objVicus_3D <- graphMatrix(X, algorithm="Vicus", ndim=3)
objLEM_3D <- graphMatrix(X, algorithm="LEM", ndim=3)
objHLLE_3D <- graphMatrix(X, algorithm="HLLE", ndim=3)
The following step is the same as in 2D Embedding case above.
outVicus_3D <- embedding(objVicus_3D)
outLEM_3D <- embedding(objLEM_3D)
outHLLE_3D <- embedding(objHLLE_3D)
layout(cbind(1:2, 3:4))
scatterplot3d(X, color=colors, pch=16, main="Original Data", angle=40)
scatterplot3d(outVicus_3D, color=colors, pch=16, main="Vicus", angle=40)
scatterplot3d(outLEM_3D, color=colors, pch=16, main="LEM", angle=70)
scatterplot3d(outHLLE_3D, color=colors, pch=16, main="HLLE", angle=70)
