geometric_kernels.kernels.product

This module provides the ProductGeometricKernel kernel for constructing product kernels from a sequence of kernels.

See this page for a brief account on theory behind product kernels and the Torus.ipynb notebook for a tutorial on how to use them.

Module Contents

class geometric_kernels.kernels.product.ProductGeometricKernel(*kernels, dimension_indices=None)

Bases: geometric_kernels.kernels.base.BaseGeometricKernel

Product kernel, defined as the product of a sequence of kernels.

See this page for a brief account on theory behind product kernels and the Torus.ipynb notebook for a tutorial on how to use them.

Parameters:

• *kernels (geometric_kernels.kernels.base.BaseGeometricKernel) – A sequence of kernels to compute the product of. Cannot contain another instance of ProductGeometricKernel. We denote the number of factors, i.e. the length of the “sequence”, by s.

• dimension_indices (beartype.typing.Optional[beartype.typing.List[beartype.typing.List[int]]]) –

  Determines how a product kernel input vector x is to be mapped into the inputs xi for the factor kernels. xi are assumed to be equal to x[dimension_indices[i]], possibly up to a reshape. Such a reshape might be necessary to accommodate the spaces whose elements are matrices rather than vectors, as determined by element_shapes. The transformation of x into the list of xis is performed by project_product().

  If None, assumes the each input is layed-out flattened and concatenated, in the same order as the factor spaces. In this case, the inverse to project_product() is make_product().

  Defaults to None.

Note

params of a ProductGeometricKernel are such that params[“lengthscale”] and params[“nu”] are (s,)-shaped arrays, where s is the number of factors.

Basically, params[“lengthscale”][i] stores the length scale parameter for the i-th factor kernel. Same goes for params[“nu”]. Importantly, this enables automatic relevance determination-like behavior.

property space: beartype.typing.List[geometric_kernels.spaces.Space]

The list of spaces upon which the factor kernels are defined.

Return type:

beartype.typing.List[geometric_kernels.spaces.Space]

K(params, X, X2=None, **kwargs)

Compute the cross-covariance matrix between two batches of vectors of inputs, or batches of matrices of inputs, depending on the space.

Parameters:

• params (beartype.typing.Dict[str, lab.Numeric]) –

  A dict of kernel parameters, typically containing two keys: “lengthscale” for length scale and “nu” for smoothness.

  The types of values in the params dict determine the output type and the backend used for the internal computations, see the warning below for more details.

  Note

  The values params[“lengthscale”] and params[“nu”] are typically (1,)-shaped arrays of the suitable backend. This serves to point at the backend to be used for internal computations.

  In some cases, for example, when the kernel is ProductGeometricKernel, the values of params may be (s,)-shaped arrays instead, where s is the number of factors.

  Note

  Finite values of params[“nu”] typically correspond to the generalized (geometric) Matérn kernels.

  Infinite params[“nu”] typically corresponds to the heat kernel (a.k.a. diffusion kernel, generalized squared exponential kernel, generalized Gaussian kernel, generalized RBF kernel). Although it is often considered to be a separate entity, we treat the heat kernel as a member of the Matérn family, with smoothness parameter equal to infinity.

• X – A batch of N inputs, each of which is a vector or a matrix, depending on how the elements of the self.space are represented.

• X2 –

  A batch of M inputs, each of which is a vector or a matrix, depending on how the elements of the self.space are represented.

  X2=None sets X2=X1.

  Defaults to None.

Returns:

The N x M cross-covariance matrix.

Return type:

lab.Numeric

Warning

The types of values in the params dict determine the backend used for internal computations and the output type.

Even if, say, geometric_kernels.jax is imported but the values in the params dict are NumPy arrays, the output type will be a NumPy array, and NumPy will be used for internal computations. To get a JAX array as an output and use JAX for internal computations, all the values in the params dict must be JAX arrays.

K_diag(params, X)

Returns the diagonal of the covariance matrix self.K(params, X, X), typically in a more efficient way than actually computing the full covariance matrix with self.K(params, X, X) and then extracting its diagonal.

Parameters:

• params – Same as for K().

• X – A batch of N inputs, each of which is a vector or a matrix, depending on how the elements of the self.space are represented.

Returns:

The N-dimensional vector representing the diagonal of the covariance matrix self.K(params, X, X).

init_params()

Returns a dict params where params[“lengthscale”] is the concatenation of all self.kernels[i].init_params()[“lengthscale”] and same for params[“nu”].

Return type:

beartype.typing.Dict[str, lab.NPNumeric]