While Principal Component Analysis (PCA) aims at minimizing global estimation error, Local Learning
Projection (LLP) approach tries to find the projection with the minimal *local*
estimation error in the sense that each projected datum can be well represented
based on ones neighbors. For the kernel part, we only enabled to use
a gaussian kernel as suggested from the original paper. The parameter `lambda`

controls possible rank-deficiency of kernel matrix.

```
do.llp(
X,
ndim = 2,
type = c("proportion", 0.1),
symmetric = c("union", "intersect", "asymmetric"),
preprocess = c("center", "scale", "cscale", "decorrelate", "whiten"),
t = 1,
lambda = 1
)
```

## Arguments

- X
an \((n\times p)\) matrix or data frame whose rows are observations

- 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.

- t
bandwidth for heat kernel in \((0,\infty)\).

- lambda
regularization parameter for kernel matrix in \([0,\infty)\).

## 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

Wu M, Yu K, Yu S, Schölkopf B (2007).
“Local Learning Projections.”
In *Proceedings of the 24th International Conference on Machine Learning*, 1039--1046.

## Examples

```
# \donttest{
## generate data
set.seed(100)
X <- aux.gensamples(n=100, dname="crown")
## test different lambda - regularization - values
out1 <- do.llp(X,ndim=2,lambda=0.1)
out2 <- do.llp(X,ndim=2,lambda=1)
out3 <- do.llp(X,ndim=2,lambda=10)
# visualize
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3))
plot(out1$Y, pch=19, main="lambda=0.1")
plot(out2$Y, pch=19, main="lambda=1")
plot(out3$Y, pch=19, main="lambda=10")
par(opar)
# }
```