Graph

GraphKernel

GraphKernel(
    laplacian: Num[Array, "N N"],
    active_dims: Union[list[int], slice, None] = None,
    lengthscale: Union[
        ScalarFloat, Float[Array, " D"], AbstractUnwrappable
    ] = 1.0,
    variance: Union[ScalarFloat, AbstractUnwrappable] = 1.0,
    smoothness: ScalarFloat = 1.0,
    n_dims: Union[int, None] = None,
    compute_engine: AbstractKernelComputation = EigenKernelComputation(),
)

Bases: StationaryKernel

The Matérn graph kernel defined on the vertex set of a graph.

A Matérn graph kernel defined through the graph Laplacian spectrum.

The kernel evaluates a Matérn spectral filter on each Laplacian eigenvalue \(\lambda\): $$ \Phi(\lambda) = \left(\frac{2\nu}{\ell^2} + \lambda\right)^{-\nu}, $$ where \(\ell\) is the lengthscale parameter and \(\nu\) is the smoothness parameter. The resulting spectral weights are normalised and scaled by the variance parameter.

The key reference for this object is Borovitskiy et al. (2021).

Parameters:

• laplacian (Num[Array, 'N N']) –

  the Laplacian matrix of the graph.

• active_dims (Union[list[int], slice, None], default: None ) –

  The indices of the input dimensions that the kernel operates on.

• lengthscale (Union[ScalarFloat, Float[Array, ' D'], AbstractUnwrappable], default: 1.0 ) –

  the lengthscale(s) of the kernel ℓ. If a scalar or an array of length 1, the kernel is isotropic, meaning that the same lengthscale is used for all input dimensions. If an array with length > 1, the kernel is anisotropic, meaning that a different lengthscale is used for each input.

• variance (Union[ScalarFloat, AbstractUnwrappable], default: 1.0 ) –

  the variance of the kernel σ.

• smoothness (ScalarFloat, default: 1.0 ) –

  the smoothness parameter of the Matérn kernel.

• n_dims (Union[int, None], default: None ) –

  The number of input dimensions. If lengthscale is an array, this argument is ignored.

• compute_engine (AbstractKernelComputation, default: EigenKernelComputation() ) –

  The computation engine that the kernel uses to compute the covariance matrix.

spectral_density property

spectral_density: Normal | StudentT

The spectral density of the kernel.

Returns:

• Normal | StudentT –

  Callable[[Float[Array, "D"]], Float[Array, "D"]]: The spectral density function.

cross_covariance

cross_covariance(
    x: Num[Array, "N D"], y: Num[Array, "M D"]
) -> Float[Array, "N M"]

Compute the cross-covariance matrix of the kernel.

Parameters:

• x (Num[Array, 'N D']) –

  the first input matrix of shape (N, D).

• y (Num[Array, 'M D']) –

  the second input matrix of shape (M, D).

Returns:

• Float[Array, 'N M'] –

  The cross-covariance matrix of the kernel of shape (N, M).

gram

gram(x: Num[Array, 'N D']) -> lx.AbstractLinearOperator

Compute the gram matrix of the kernel.

Parameters:

• x (Num[Array, 'N D']) –

  the input matrix of shape (N, D).

Returns:

• AbstractLinearOperator –

  The gram matrix of the kernel of shape (N, N).

diagonal

diagonal(x: Num[Array, 'N D']) -> lx.AbstractLinearOperator

Compute the diagonal of the gram matrix of the kernel.

Parameters:

• x (Num[Array, 'N D']) –

  the input matrix of shape (N, D).

Returns:

• AbstractLinearOperator –

  The diagonal of the gram matrix of the kernel of shape (N,).

slice_input

slice_input(
    x: Float[Array, "... D"],
) -> Float[Array, "... Q"]

Slice out the relevant columns of the input matrix.

Select the relevant columns of the supplied matrix to be used within the kernel's evaluation.

Parameters:

• x (Float[Array, '... D']) –

  the matrix or vector that is to be sliced.

Returns:

• Float[Array, '... Q'] –

  The sliced form of the input matrix.

__add__

__add__(
    other: Union[AbstractKernel, ScalarFloat],
) -> AbstractKernel

Add two kernels together. Args: other (AbstractKernel): The kernel to be added to the current kernel.

Returns:

• AbstractKernel ( AbstractKernel ) –

  A new kernel that is the sum of the two kernels.

__mul__

__mul__(
    other: Union[AbstractKernel, ScalarFloat],
) -> AbstractKernel

Multiply two kernels together.

Parameters:

• other (AbstractKernel) –

  The kernel to be multiplied with the current kernel.

Returns:

• AbstractKernel ( AbstractKernel ) –

  A new kernel that is the product of the two kernels.