inchol function

Incomplete Cholesky decomposition

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

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.

Returns

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))

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

Author(s)

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

alexandros.karatzoglou@ci.tuwien.ac.at

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,]