shrinkGPR (version 1.0.0)

kernel_functions: Kernel Functions for Gaussian Processes

Description

A set of kernel functions for Gaussian processes, including the squared exponential (SE) kernel and Matérn kernels with smoothness parameters 1/2, 3/2, and 5/2. These kernels compute the covariance structure for Gaussian process regression models and are designed for compatibility with the shrinkGPR function.

Usage

kernel_se(thetas, tau, x, x_star = NULL)
kernel_matern_12(thetas, tau, x, x_star = NULL)
kernel_matern_32(thetas, tau, x, x_star = NULL)
kernel_matern_52(thetas, tau, x, x_star = NULL)

Value

A torch_tensor containing the batched covariance matrices (one for each latent sample):

• If x_star = NULL, the output is of dimensions n_latent x N x N, representing pairwise covariances between all points in x.

• If x_star is provided, the output is of dimensions n_latent x N_new x N, representing pairwise covariances between x_star and x.

Arguments

thetas

A torch_tensor of dimensions n_latent x d, representing the latent length-scale parameters.

tau

A torch_tensor of length n_latent, representing the latent scaling factors.

x

A torch_tensor of dimensions N x d, containing the input data points.

x_star

Either NULL or a torch_tensor of dimensions N_new x d. If NULL, the kernel is computed for x against itself. Otherwise, it computes the kernel between x and x_star.

Details

These kernel functions are used to define the covariance structure in Gaussian process regression models. Each kernel implements a specific covariance function:

• kernel_se: Squared exponential (SE) kernel, also known as the radial basis function (RBF) kernel. It assumes smooth underlying functions.

• kernel_matern_12: Matérn kernel with smoothness parameter \(\nu = 1/2\), equivalent to the absolute exponential kernel.

• kernel_matern_32: Matérn kernel with smoothness parameter \(\nu = 3/2\).

• kernel_matern_52: Matérn kernel with smoothness parameter \(\nu = 5/2\).

The sqdist helper function is used internally by these kernels to compute squared distances between data points.

Note that these functions perform no input checks, as to ensure higher performance. Users should ensure that the input tensors are of the correct dimensions.

Examples

if (torch::torch_is_installed()) {
  # Example inputs
  torch::torch_manual_seed(123)
  n_latent <- 3
  d <- 2
  N <- 5
  thetas <- torch::torch_randn(n_latent, d)$abs()
  tau <- torch::torch_randn(n_latent)$abs()
  x <- torch::torch_randn(N, d)

  # Compute the SE kernel
  K_se <- kernel_se(thetas, tau, x)
  print(K_se)

  # Compute the Matérn 3/2 kernel
  K_matern32 <- kernel_matern_32(thetas, tau, x)
  print(K_matern32)

  # Compute the Matérn 5/2 kernel with x_star
  x_star <- torch::torch_randn(3, d)
  K_matern52 <- kernel_matern_52(thetas, tau, x, x_star)
  print(K_matern52)
}