Local Discriminant Embedding (LDE) suffers from a small-sample-size problem where
scatter matrix may suffer from rank deficiency. Exponential LDE (ELDE) provides
not only a remedy for the problem using matrix exponential, but also a flexible
framework to transform original data into a new space via distance diffusion mapping
similar to kernel-based nonlinear mapping.

```
do.elde(
X,
label,
ndim = 2,
t = 1,
preprocess = c("center", "scale", "cscale", "decorrelate", "whiten"),
k1 = max(ceiling(nrow(X)/10), 2),
k2 = max(ceiling(nrow(X)/10), 2)
)
```

## Arguments

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

- label
a length-\(n\) vector of data class labels.

- ndim
an integer-valued target dimension.

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

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

for more details.

- k1
the number of same-class neighboring points (homogeneous neighbors).

- k2
the number of different-class neighboring points (heterogeneous neighbors).

## 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

Dornaika F, Bosaghzadeh A (2013).
“Exponential Local Discriminant Embedding and Its Application to Face Recognition.”
*IEEE Transactions on Cybernetics*, **43**(3), 921--934.

## Examples

```
## generate data of 3 types with difference
set.seed(100)
dt1 = aux.gensamples(n=20)-50
dt2 = aux.gensamples(n=20)
dt3 = aux.gensamples(n=20)+50
## merge the data and create a label correspondingly
X = rbind(dt1,dt2,dt3)
label = rep(1:3, each=20)
## try different kernel bandwidth
out1 = do.elde(X, label, t=1)
out2 = do.elde(X, label, t=10)
out3 = do.elde(X, label, t=100)
## visualize
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3))
plot(out1$Y, pch=19, col=label, main="ELDE::bandwidth=1")
plot(out2$Y, pch=19, col=label, main="ELDE::bandwidth=10")
plot(out3$Y, pch=19, col=label, main="ELDE::bandwidth=100")
par(opar)
```