Rdimtools (version 1.1.2)

do.rndproj: Random Projection

Description

do.rndproj is a linear dimensionality reduction method based on random projection technique, featured by the celebrated Johnson–Lindenstrauss lemma.

Usage

do.rndproj(
  X,
  ndim = 2,
  preprocess = c("null", "center", "scale", "cscale", "whiten", "decorrelate"),
  type = c("gaussian", "achlioptas", "sparse"),
  s = max(sqrt(ncol(X)), 3)
)

Value

a named list containing

Y

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

projection

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

epsilon

an estimated error \(\epsilon\) in accordance with JL lemma.

trfinfo

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

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.

preprocess

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

type

a type of random projection, one of "gaussian","achlioptas" or "sparse".

s

a tuning parameter for determining values in projection matrix. While default is to use \(max(log \sqrt{p},3)\), it is required for \(s \ge 3\).

Details

The Johnson-Lindenstrauss(JL) lemma states that given \(0 < \epsilon < 1\), for a set \(X\) of \(m\) points in \(R^N\) and a number \(n > 8log(m)/\epsilon^2\), there is a linear map \(f:R^N\) to R^n such that $$(1-\epsilon)|u-v|^2 \le |f(u)-f(v)|^2 \le (1+\epsilon)|u-v|^2$$ for all \(u,v\) in \(X\).

Three types of random projections are supported for an (p-by-ndim) projection matrix \(R\).

1. Conventional approach is to use normalized Gaussian random vectors sampled from unit sphere \(S^{p-1}\).

2. Achlioptas suggested to employ a sparse approach using samples from \(\sqrt{3}(1,0,-1)\) with probability \((1/6,4/6,1/6)\).

3. Li et al proposed to sample from \(\sqrt{s}(1,0,-1)\) with probability \((1/2s,1-1/s,1/2s)\) for \(s\ge 3\) to incorporate sparsity while attaining speedup with little loss in accuracy. While the original suggsetion from the authors is to use \(\sqrt{p}\) or \(p/log(p)\) for \(s\), any user-supported \(s \ge 3\) is allowed.

References

johnson_extensions_1984Rdimtools

achlioptas_databasefriendly_2003Rdimtools

li_very_2006Rdimtools

Examples

## use iris data
data(iris)
set.seed(100)
subid = sample(1:150, 50)
X     = as.matrix(iris[subid,1:4])
label = as.factor(iris[subid,5])

## 1. Gaussian projection
output1 <- do.rndproj(X,ndim=2)

## 2. Achlioptas projection
output2 <- do.rndproj(X,ndim=2,type="achlioptas")

## 3. Sparse projection
output3 <- do.rndproj(X,type="sparse")

## Visualize three different projections
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3))
plot(output1$Y, pch=19, col=label, main="RNDPROJ::Gaussian")
plot(output2$Y, pch=19, col=label, main="RNDPROJ::Arclioptas")
plot(output3$Y, pch=19, col=label, main="RNDPROJ::Sparse")
par(opar)