# Kernel Functions for Gaussian Processes 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. ```r 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) ``` ## 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`. ## Returns 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`. ## 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 ```r 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) } ```

kernel_functions function