Describes the kernel matrix \(K\) with entries \(k(x_i, x_j)\) without computing it. Randomly pivoted Cholesky then evaluates only the entries it needs, roughly \((k + 1) n\) of the \(n^2\), which is what makes it practical when the full kernel matrix would not fit in memory.
Usage
kernel_matrix(
X,
kernel = c("gaussian", "laplace", "matern32", "matern52"),
bandwidth = NULL,
block = 1000L
)Arguments
- X
Numeric matrix with one row per point.
- kernel
Kernel family:
"gaussian"\(\exp(-r^2 / 2h^2)\),"laplace"\(\exp(-r / h)\), or the Matern kernels"matern32"and"matern52", where \(r\) is the Euclidean distance.- bandwidth
Length scale \(h\). Defaults to the median distance between points, computed on at most the first 500 rows.
- block
Number of rows computed at a time when multiplying by the whole matrix, which bounds memory use.
Value
An object of class matsketch_kernel, accepted wherever
matsketch expects a matrix. as.matrix() forms the full matrix.
Details
Each product with the whole matrix recomputes the kernel, block by block.
For methods that multiply many times, such as pcg(), it is faster to
form the matrix once with as.matrix() whenever it fits in memory.
Examples
X <- matrix(rnorm(400), ncol = 2)
K <- kernel_matrix(X, "gaussian")
K
#> <matsketch_kernel> gaussian kernel on 200 points, bandwidth 1.632
#> full matrix would hold 40,000 entries; nothing is stored
K$block(1:3, 1:3)
#> [,1] [,2] [,3]
#> [1,] 1.0000000 0.7960616 0.8923518
#> [2,] 0.7960616 1.0000000 0.5203043
#> [3,] 0.8923518 0.5203043 1.0000000