Locally Linear Embedding

Locally-Linear Embedding (LLE) was introduced approximately at the same time as Isomap. Its idea was motivated to describe entire data manifold by making a chain of local patches in that low-dimensional embedding should resemble the connectivity pattern of patches. do.lle also provides an automatic choice of regularization parameter based on an optimality criterion suggested by authors.

do.lle(
  X,
  ndim = 2,
  type = c("proportion", 0.1),
  symmetric = "union",
  weight = TRUE,
  preprocess = c("null", "center", "scale", "cscale", "decorrelate", "whiten"),
  regtype = FALSE,
  regparam = 1
)

Arguments

X

an \((n\times p)\) matrix or data frame whose rows are observations and columns represent independent variables.

ndim

an integer-valued target dimension.

type

a vector of neighborhood graph construction. Following types are supported; c("knn",k), c("enn",radius), and c("proportion",ratio). Default is c("proportion",0.1), connecting about 1/10 of nearest data points among all data points. See also aux.graphnbd for more details.

symmetric

one of "intersect", "union" or "asymmetric" is supported. Default is "union". See also aux.graphnbd for more details.

weight

TRUE to perform LLE on weighted graph, or FALSE otherwise.

preprocess

an additional option for preprocessing the data. Default is "null". See also aux.preprocess for more details.

regtype

TRUE for automatic regularization parameter selection, FALSE otherwise as default.

regparam

regularization parameter.

Value

a named list containing

Y

an \((n\times ndim)\) matrix whose rows are embedded observations.

trfinfo

a list containing information for out-of-sample prediction.

eigvals

a vector of eigenvalues from computation of embedding matrix.

References

Roweis ST (2000). “Nonlinear Dimensionality Reduction by Locally Linear Embedding.” Science, 290(5500), 2323--2326.

Author

Kisung You

Examples

# \donttest{
## generate swiss-roll data
set.seed(100)
X = aux.gensamples(n=100)

## 1. connecting 10% of data for graph construction.
output1 <- do.lle(X,ndim=2,type=c("proportion",0.10))

## 2. constructing 20%-connected graph
output2 <- do.lle(X,ndim=2,type=c("proportion",0.20))

## 3. constructing 50%-connected with bigger regularization parameter
output3 <- do.lle(X,ndim=2,type=c("proportion",0.5),regparam=10)

## Visualize three different projections
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3))
plot(output1$Y, main="5%")
plot(output2$Y, main="10%")
plot(output3$Y, main="50%+Binary")

par(opar)
# }