Locally Linear Embedding (LLE) is a powerful nonlinear manifold learning method. This method,
Locally Linear Embedded Eigenspace Analysis - LEA, in short - is a linear approximation to LLE,
similar to Neighborhood Preserving Embedding. In our implementation, the choice of weight binarization
is removed in order to respect original work. For 1-dimensional projection, which is rarely performed,
authors provided a detour for rank correcting mechanism but it is omitted for practical reason.

```
do.lea(
X,
ndim = 2,
type = c("proportion", 0.1),
symmetric = c("union", "intersect", "asymmetric"),
preprocess = c("center", "scale", "cscale", "decorrelate", "whiten")
)
```

## 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.

- preprocess
an additional option for preprocessing the data.
Default is "center". See also `aux.preprocess`

for more details.

## 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.

- projection
a \((p\times ndim)\) whose columns are basis for projection.

## References

Fu Y, Huang TS (2005).
“Locally Linear Embedded Eigenspace Analysis.”
*IFP-TR, UIUC*, **2005**, 2--05.

## Examples

```
if (FALSE) {
## use iris dataset
data(iris)
set.seed(100)
subid <- sample(1:150, 50)
X <- as.matrix(iris[subid,1:4])
lab <- as.factor(iris[subid,5])
## compare LEA with LLE and another approximation NPE
out1 <- do.lle(X, ndim=2)
out2 <- do.npe(X, ndim=2)
out3 <- do.lea(X, ndim=2)
## visual comparison
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3))
plot(out1$Y, pch=19, col=lab, main="LLE")
plot(out2$Y, pch=19, col=lab, main="NPE")
plot(out3$Y, pch=19, col=lab, main="LEA")
par(opar)
}
```