Local Learning Projections

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)
# }