Basis Functions

gpjax.kernels.computations.basis_functions

Kernel = tp.TypeVar('Kernel', bound='gpjax.kernels.base.AbstractKernel') module-attribute

BasisFunctionComputation dataclass

Bases: AbstractKernelComputation

Compute engine class for finite basis function approximations to a kernel.

diagonal(kernel: Kernel, inputs: Num[Array, 'N D']) -> Diagonal

For a given kernel, compute the elementwise diagonal of the NxN gram matrix on an input matrix of shape NxD.

Parameters:

Name	Type	Description	Default
kernel	AbstractKernel	the kernel function.	required
inputs	Float[Array, 'N D']	The input matrix.	required

Returns

Diagonal: The computed diagonal variance entries.

cross_covariance(kernel: Kernel, x: Float[Array, 'N D'], y: Float[Array, 'M D']) -> Float[Array, 'N M']

Compute an approximate cross-covariance matrix.

For a pair of inputs, compute the cross covariance matrix between the inputs.

Parameters:

Name	Type	Description	Default
kernel	Kernel	the kernel function.	required
x	Float[Array, 'N D']	(Float[Array, "N D"]): A $N \times D$ array of inputs.	required
y	Float[Array, 'M D']	(Float[Array, "M D"]): A $M \times D$ array of inputs.	required

Returns:

Type	Description
Float[Array, 'N M']	Float[Array, "N M"]: A $N \times M$ array of cross-covariances.

gram(kernel: Kernel, inputs: Float[Array, 'N D']) -> LinearOperator

Compute an approximate Gram matrix.

For the Gram matrix, we can save computations by computing only one matrix multiplication between the inputs and the scaled frequencies.

Parameters:

Name	Type	Description	Default
kernel	Kernel	the kernel function.	required
inputs	Float[Array, 'N D']	A $N x D$ array of inputs.	required

Returns:

Name	Type	Description
LinearOperator	LinearOperator	A dense linear operator representing the $N \times N$ Gram matrix.

compute_features(kernel: Kernel, x: Float[Array, 'N D']) -> Float[Array, 'N L']

Compute the features for the inputs.

Parameters:

Name	Type	Description	Default
kernel	Kernel	the kernel function.	required
x	Float[Array, 'N D']	A $N \times D$ array of inputs.	required

Returns

Float[Array, "N L"]: A $N \times L$ array of features where $L = 2M$.

scaling(kernel: Kernel)

Compute the scaling factor for the covariance matrix.

Parameters:

Name	Type	Description	Default
kernel	Kernel	the kernel function.	required

Returns

Float[Array, ""]: A scalar array.