The csi function in kernlab is an implementation of an
incomplete Cholesky decomposition algorithm which exploits side
information (e.g., classification labels, regression responses) to
compute a low rank decomposition of a kernel matrix from the data.
# S4 method for matrix
csi(x, y, kernel="rbfdot", kpar=list(sigma=0.1), rank,
centering = TRUE, kappa = 0.99 ,delta = 40 ,tol = 1e-5)The data matrix indexed by row
the classification labels or regression responses. In classification y is a \(m \times n\) matrix where \(m\) the number of data and \(n\) the number of classes \(y\) and \(y_i\) is 1 if the corresponding x belongs to class i.
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
stringdot String kernel
The kernel parameter can also be set to a user defined function of class kernel by passing the function name as an argument.
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.
maximal rank of the computed kernel matrix
if TRUE centering is performed (default: TRUE)
trade-off between approximation of K and prediction of Y (default: 0.99)
number of columns of cholesky performed in advance (default: 40)
minimum gain at each iteration (default: 1e-4)
An S4 object of class "csi" which is an extension of the class "matrix". The object is the decomposed kernel matrix along with the slots :
Indices on which pivots where done
Residuals left on the diagonal
Residuals picked for pivoting
predicted gain before adding each column
actual gain after adding each column
QR decomposition of the kernel matrix
QR decomposition of the kernel matrix
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.
csi uses the class labels, or regression responses to compute a
more appropriate approximation for the problem at hand considering the
additional information from the response variable.
Francis R. Bach, Michael I. Jordan Predictive low-rank decomposition for kernel methods. Proceedings of the Twenty-second International Conference on Machine Learning (ICML) 2005 http://www.di.ens.fr/~fbach/bach_jordan_csi.pdf
# NOT RUN {
data(iris)
## create multidimensional y matrix
yind <- t(matrix(1:3,3,150))
ymat <- matrix(0, 150, 3)
ymat[yind==as.integer(iris[,5])] <- 1
datamatrix <- as.matrix(iris[,-5])
# initialize kernel function
rbf <- rbfdot(sigma=0.1)
rbf
Z <- csi(datamatrix,ymat, kernel=rbf, rank = 30)
dim(Z)
pivots(Z)
# calculate kernel matrix
K <- crossprod(t(Z))
# difference between approximated and real kernel matrix
(K - kernelMatrix(kernel=rbf, datamatrix))[6,]
# }
Run the code above in your browser using DataLab