kernlab (version 0.9-33)

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)

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))

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 kernel functions which can be used by setting the kernel parameter to the following strings:

• rbfdot Radial Basis kernel function "Gaussian"

• polydot Polynomial kernel function

• vanilladot Linear kernel function

• tanhdot Hyperbolic tangent kernel function

• laplacedot Laplacian kernel function

• besseldot Bessel kernel function

• anovadot ANOVA RBF kernel function

• splinedot Spline kernel

The kernel parameter can also be set to a user defined function of class kernel by passing the function name as an argument.

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 :

• sigma inverse kernel width for the Radial Basis kernel function "rbfdot" and the Laplacian kernel "laplacedot".

• degree, scale, offset for the Polynomial kernel "polydot"

• scale, offset for the Hyperbolic tangent kernel function "tanhdot"

• sigma, order, degree for the Bessel kernel "besseldot".

• sigma, degree for the ANOVA kernel "anovadot".

Hyper-parameters for user defined kernels can be passed through the kpar parameter as well.

tol

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

maxiter

maximum number of iterations and columns in \(Z\)

blocksize

add this many columns to matrix per iteration

verbose

print info on algorithm convergence

Author

Alexandros Karatzoglou (based on Matlab code by S.V.N. (Vishy) Vishwanathan and Alex Smola)

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
https://www.jmlr.org/papers/volume3/bach02a/bach02a.pdf

See Also

csi, inchol-class, chol

Examples

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,]