KERE (version 1.0.0)

kernelMatrix: Kernel Matrix functions

Description

kernelMatrix calculates the kernel matrix $K_{ij} = k(x_i,x_j)$ or $K_{ij} = k(x_i,y_j)$. kernelPol computes the quadratic kernel expression $H = z_i z_j k(x_i,x_j)$, $H = z_i k_j k(x_i,y_j)$. kernelMult calculates the kernel expansion $f(x_i) = \sum_{i=1}^m z_i k(x_i,x_j)$ kernelFast computes the kernel matrix, identical to kernelMatrix, except that it also requires the squared norm of the first argument as additional input, useful in iterative kernel matrix calculations.

Usage

## S3 method for class 'kernel':
kernelMatrix(kernel, x, y = NULL)
## S3 method for class 'kernel':
kernelPol(kernel, x, y = NULL, z, k = NULL)
## S3 method for class 'kernel':
kernelMult(kernel, x, y = NULL, z, blocksize = 256)
## S3 method for class 'kernel':
kernelFast(kernel, x, y, a)

Arguments

kernel

the kernel function to be used to calculate the kernel matrix. This has to be a function of class kernel, i.e. which can be generated either one of the build in kernel generating functions (e.g., rbfdot etc.) or a user

x

a data matrix to be used to calculate the kernel matrix.

y

second data matrix to calculate the kernel matrix.

z

a suitable vector or matrix

k

a suitable vector or matrix

a

the squared norm of x, e.g., rowSums(x^2)

blocksize

the kernel expansion computations are done block wise to avoid storing the kernel matrix into memory. blocksize defines the size of the computational blocks.

Value

• kernelMatrix returns a symmetric diagonal semi-definite matrix. kernelPol returns a matrix. kernelMult usually returns a one-column matrix.

Details

Common functions used during kernel based computations. The kernel parameter can be set to any function, of class kernel, which computes the inner product in feature space between two vector arguments. KERE provides the most popular kernel functions which can be initialized by using the following functions:

• rbfdotRadial Basis kernel function
• polydotPolynomial kernel function
• vanilladotLinear kernel function
• tanhdotHyperbolic tangent kernel function
• laplacedotLaplacian kernel function
• besseldotBessel kernel function
• anovadotANOVA RBF kernel function
• splinedotthe Spline kernel

(see example.)

kernelFast is mainly used in situations where columns of the kernel matrix are computed per invocation. In these cases, evaluating the norm of each row-entry over and over again would cause significant computational overhead.

See Also

rbfdot, polydot, tanhdot, vanilladot

Examples

## use the spam data
x <- matrix(rnorm(10*10),10,10)

## initialize kernel function 
rbf <- rbfdot(sigma = 0.05)
rbf

## calculate kernel matrix
kernelMatrix(rbf, x)

y <- matrix(rnorm(10*1),10,1)


## calculate the quadratic kernel expression
kernelPol(rbf, x, ,y)

## calculate the kernel expansion
kernelMult(rbf, x, ,y)