Although the method of Kernel Principal Component Analysis (KPCA) was originally developed to visualize non-linearly distributed data in Euclidean space, we graft this to the case for manifolds where extrinsic geometry is explicitly available. The algorithm uses Gaussian kernel with $$K(X_i, X_j) = \exp\left( - \frac{d^2 (X_i, X_j)}{2 \sigma^2} \right )$$ where \(\sigma\) is a bandwidth parameter and \(d(\cdot, \cdot)\) is an extrinsic distance defined on a specific manifold.
Usage
riem.kpca(riemobj, ndim = 2, sigma = 1, negative = c("truncate", "error"))Arguments
- riemobj
a S3
"riemdata"class for \(N\) manifold-valued data.- ndim
an integer-valued target dimension (default: 2).
- sigma
A finite, strictly positive Gaussian bandwidth.
- negative
Treatment of an empirically indefinite kernel:
"truncate"warns and returns a positive-spectrum embedding;"error"stops.
Value
a named list containing
- embed
an \((N\times ndim)\) matrix whose rows are embedded observations.
- vars
a length-\(N\) vector of eigenvalues from kernelized covariance matrix.
Details
Scores are eigenvectors multiplied by square roots of the centered Gram eigenvalues. They are ordered by decreasing eigenvalue. A Gaussian of an arbitrary manifold dissimilarity is not automatically a positive-definite kernel. For an indefinite Gram matrix, truncated coordinates are an approximation, not exact kernel principal component scores. The full spectrum, negative eigenvalues, and empirical kernel status are returned. This function remains a training-data embedding, without a predict method.
References
Schölkopf B, Smola A, Müller K (1997). “Kernel principal component analysis.” In Goos G, Hartmanis J, van Leeuwen J, Gerstner W, Germond A, Hasler M, Nicoud J (eds.), Artificial Neural Networks — ICANN'97, volume 1327, 583–588. Springer Berlin Heidelberg, Berlin, Heidelberg. ISBN 978-3-540-63631-1 978-3-540-69620-9.
Examples
#-------------------------------------------------------------------
# Example for Gorilla Skull Data : 'gorilla'
#-------------------------------------------------------------------
## PREPARE THE DATA
# Aggregate two classes into one set
data(gorilla)
mygorilla = array(0,c(8,2,59))
for (i in 1:29){
mygorilla[,,i] = gorilla$male[,,i]
}
for (i in 30:59){
mygorilla[,,i] = gorilla$female[,,i-29]
}
gor.riem = wrap.landmark(mygorilla)
gor.labs = c(rep("red",29), rep("blue",30))
## APPLY KPCA WITH DIFFERENT KERNEL BANDWIDTHS
kpca1 = riem.kpca(gor.riem, sigma=0.01)
kpca2 = riem.kpca(gor.riem, sigma=1)
#> Warning: The Gaussian dissimilarity kernel is indefinite; returning an explicitly truncated positive-spectrum embedding.
kpca3 = riem.kpca(gor.riem, sigma=100)
#> Warning: The Gaussian dissimilarity kernel is indefinite; returning an explicitly truncated positive-spectrum embedding.
## VISUALIZE
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3), pty="s")
plot(kpca1$embed, pch=19, col=gor.labs, main="sigma=1/100")
plot(kpca2$embed, pch=19, col=gor.labs, main="sigma=1")
plot(kpca3$embed, pch=19, col=gor.labs, main="sigma=100")
par(opar)