Polynomial

Polynomial

Polynomial(active_dims=None, degree=2, shift=0.0, variance=1.0, n_dims=None, compute_engine=DenseKernelComputation())

Bases: AbstractKernel

The Polynomial kernel with variable degree.

Computes the covariance for pairs of inputs \((x, y)\) with variance \(\sigma^2\): $$ k(x, y) = (\alpha + \sigma^2 x y)^d $$ where \(\sigma^\in \mathbb{R}_{>0}\) is the kernel's variance parameter, shift parameter \(\alpha\) and integer degree \(d\).

Parameters:

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

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

• degree (int, default: 2 ) –

  The degree of the polynomial.

• shift (Union[ScalarFloat, Variable[ScalarArray]], default: 0.0 ) –

  The shift parameter of the kernel.

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

  The variance of the kernel.

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

  The number of input dimensions.

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

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

cross_covariance

cross_covariance(x, y)

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)

Compute the gram matrix of the kernel.

Parameters:

• x (Num[Array, 'N D']) –

  the input matrix of shape (N, D).

Returns:

• LinearOperator –

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

diagonal

diagonal(x)

Compute the diagonal of the gram matrix of the kernel.

Parameters:

• x (Num[Array, 'N D']) –

  the input matrix of shape (N, D).

Returns:

• Float[Array, ' N'] –

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

slice_input

slice_input(x)

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)

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)

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.