# riem.kpca: Kernel Principal Component Analysis In Riemann: Learning with Data on Riemannian Manifolds

 riem.kpca R Documentation

## Kernel Principal Component Analysis

### Description

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≤ft( - \frac{d^2 (X_i, X_j)}{2 σ^2} \right )

where σ 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)


### Arguments

 riemobj a S3 "riemdata" class for N manifold-valued data. ndim an integer-valued target dimension (default: 2). sigma the bandwidth parameter (default: 1).

### 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.

### References

\insertRef

scholkopf_kernel_1997Riemann

### 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)
kpca3 = riem.kpca(gor.riem, sigma=100)
## VISUALIZE
plot(kpca1$embed, pch=19, col=gor.labs, main="sigma=1/100") plot(kpca2$embed, pch=19, col=gor.labs, main="sigma=1")