Kernel specification

RxGP builds on KernelFunctions.jl and provides utilities for constructing parameterised kernels, computing their gradient and Hessian operators, and initialising hyperparameters.

Building a kernel

RxGP.get_simple_kernel_and_params — Function

get_simple_kernel_and_params(D; kernel_spec=:SE, num_SE=1, num_SM=1, independent_SE_lengthscales=true)

Return (kernel_fn, θ_init, dim_θ) for a parameterised kernel in input dimension D.

Supported kernel_spec values: :SE, :SEn, :SMn, :SEn_SMn.

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This function returns a tuple (kernel_fn, θ_init, dim_θ) where:

kernel_fn(θ) returns a KernelFunctions.Kernel instance parameterised by θ.
θ_init is a vector of initial hyperparameters (in softplus-transformed space).
dim_θ is the total number of hyperparameters.

Supported kernel families:

`kernel_spec`	Description	Parameters per component
`:SE`	Single squared-exponential	`1 + D` (variance + lengthscales) or `2` (shared lengthscale)
`:SEn`	Sum of `num_SE` SE kernels	`num_SE × (1 + D)` or `num_SE × 2`
`:SMn`	Sum of `num_SM` spectral-mixture components	`num_SM × (1 + 2D)`
`:SEn_SMn`	Sum of SE and SM components	Combined

The independent_SE_lengthscales flag controls whether each input dimension gets its own lengthscale (true, default) or all dimensions share one (false).

Gradient and Hessian kernel functions

For derivative observation models (:grad and :joint_fn_grad operators), RxGP needs the gradient and Hessian of the kernel with respect to the input:

RxGP.get_gradient_Kxu_fn — Function

get_gradient_Kxu_fn(D; kernel, kernel_spec=:SE, mode=:AD, independent_SE_lengthscales=true)

Return a function (x, θ, Xu) -> Matrix that computes the gradient of the cross-kernel $\nabla_x k(x, X_u)$ (a D×M matrix). Supports :AD (autodiff) and :AN (analytic, SE only) modes.

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RxGP.get_gradient_Kxx_fn — Function

get_gradient_Kxx_fn(D; kernel, kernel_spec=:SE, mode=:AD, independent_SE_lengthscales=true)

Return a function (x, θ) -> Matrix that computes the Hessian of the auto-kernel $\nabla_x \nabla_{x'} k(x, x')$ (a D×D matrix). Supports :AD and :AN modes.

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These are called internally by get_UniSGPMeta when operator is :grad or :joint_fn_grad. You only need to call them directly for advanced custom setups.

Modes:

Mode	Description	Supported kernels
`:AD`	Automatic differentiation via ForwardDiff.jl	All kernel families
`:AN`	Analytic closed-form expressions (~10× faster)	`:SE`, `:SEn` only