inchol {kernlab} | R Documentation

## 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 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

### 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.

### 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))`

### Author(s)

Alexandros Karatzoglou (based on Matlab code by S.V.N. (Vishy) Vishwanathan and Alex Smola)
alexandros.karatzoglou@ci.tuwien.ac.at

### 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

### 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,]

[Package kernlab version 0.9-27 Index]