Basis Functions

BasisFunctionComputation

Bases: AbstractKernelComputation

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

diagonal

diagonal(
    kernel: K, inputs: Float[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:

• kernel (AbstractKernel) –

  the kernel function.

• inputs (Float[Array, 'N D']) –

  The input matrix.

Returns

Diagonal: The computed diagonal variance entries.

compute_features

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

Compute the features for the inputs.

Parameters:

• kernel (K) –

  the kernel function.

• x (Float[Array, 'N D']) –

  the inputs to the kernel function of shape (N, D).

Returns:

• Float[Array, 'N L'] –

  A matrix of shape \(N \times L\) representing the random fourier features where \(L = 2M\).

scaling

scaling(kernel: K) -> Float[Array, '']

Compute the scaling factor for the covariance matrix.

Parameters:

• kernel (K) –

  the kernel function.

Returns:

• Float[Array, ''] –

  A scalar array representing the scaling factor.

gram

gram(kernel: K, x: Num[Array, 'N D']) -> Dense

For a given kernel, compute Gram covariance operator of the kernel function on an input matrix of shape (N, D).

Parameters:

• kernel (K) –

  the kernel function.

• x (Num[Array, 'N D']) –

  the inputs to the kernel function of shape (N, D).

Returns:

• Dense –

  The Gram covariance of the kernel function as a linear operator.

cross_covariance

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

For a given kernel, compute the cross-covariance matrix on an a pair of input matrices with shape (N, D) and (M, D).

Parameters:

• kernel (K) –

  the kernel function.

• 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 computed cross-covariance of shape (N, M).