kernlab (version 0.9-7)

inchol: Incomplete Cholesky decomposition

Description

inchol computes the incomplete Cholesky decomposition of the kernel matrix from a data matrix.

Usage

inchol(x, kernel="rbfdot", kpar=list(sigma=0.1), tol = 0.001, 
            maxiter = dim(x)[1], blocksize = 50, verbose = 0)

Arguments

x

The data matrix indexed by row

kernel

the kernel function used in training and predicting. This parameter can be set to any function, of class kernel, which computes the inner product in feature space between two vector arguments. kernlab provides the most popular ker

kpar

the list of hyper-parameters (kernel parameters). This is a list which contains the parameters to be used with the kernel function. Valid parameters for existing kernels are :

• sigmainverse kernel width for the Radial Basis

tol

algorithm stops when remaining pivots bring less accuracy then tol (default: 0.001)

maxiter

maximum number of iterations and colums in $Z$

blocksize

add this many columns to matrix per iteration

verbose

print info on algorithm convergence

Value

• An S4 object of class "inchol" which is an extension of the class "matrix". The object is the decomposed kernel matrix along with the slots :
• pivots
  Indices on which pivots where done
• diagresidues
  Residuals left on the diagonal
• maxresiduals
  Residuals picked for pivoting
• slots can be accessed either by object@slot or by accessor functions with the same name (e.g., pivots(object))

Details

An incomplete cholesky decomposition calculates $Z$ where $K= ZZ'$ $K$ being the kernel matrix. Since the rank of a kernel matrix is usually low, $Z$ tends to be smaller then the complete kernel matrix. The decomposed matrix can be used to create memory efficient kernel-based algorithms without the need to compute and store a complete kernel matrix in memory.

References

Francis R. Bach, Michael I. Jordan Kernel Independent Component Analysis Journal of Machine Learning Research 3, 1-48 http://www.jmlr.org/papers/volume3/bach02a/bach02a.pdf [object Object],[object Object],[object Object],[object Object]

code{csi}, code{inchol-class}, chol data(iris) datamatrix <- as.matrix(iris[,-5]) # initialize kernel function rbf <- rbfdot(sigma=0.1) rbf Z <- inchol(datamatrix,kernel=rbf) dim(Z) pivots(Z) # calculate kernel matrix K <- crossprod(t(Z)) # difference between approximated and real kernel matrix (K - kernelMatrix(kernel=rbf, datamatrix))[6,] methods algebra array