geometric_kernels.kernels.feature_map

This module provides the MaternFeatureMapKernel kernel, the basic kernel for non-compact symmetric spaces, subclasses of NoncompactSymmetricSpace.

Module Contents

class geometric_kernels.kernels.feature_map.MaternFeatureMapKernel(space, feature_map, key, normalize=True)

Bases: geometric_kernels.kernels.base.BaseGeometricKernel

This class computes a (Matérn) kernel based on a feature map.

\[k_{\nu, \kappa}(x, y) = \langle \phi_{\nu, \kappa}(x), \phi_{\nu, \kappa}(y) \rangle_{\mathbb{R}^n}\]

where \(\langle \cdot , \cdot \rangle_{\mathbb{R}^n}\) is the standard inner product in \(\mathbb{R}^n\) and \(\phi_{\nu, \kappa}: X \to \mathbb{R}^n\) is an arbitrary function called feature map. We assume that it depends on the smoothness and length scale parameters \(\nu\) and \(\kappa\), respectively, which makes this kernel specifically Matérn.

Note

A brief introduction into feature maps and related kernels can be found on this page.

Note that the finite-dimensional feature maps this kernel is meant to be used with are, in most cases, some approximations of the intractable infinite-dimensional feature maps.

Parameters:

• space (geometric_kernels.spaces.base.Space) – The space on which the kernel is defined.

• feature_map (geometric_kernels.feature_maps.FeatureMap) – A FeatureMap object that represents an arbitrary function \(\phi_{\nu, \kappa}: X \to \mathbb{R}^n\), where \(X\) is the space, \(n\) can be an arbitrary finite integer, and \(\nu, \kappa\) are the smoothness and length scale parameters.

• key (lab.RandomState) –

  Random state, either np.random.RandomState, tf.random.Generator, torch.Generator or jax.tensor (which represents a random state).

  Many feature maps used in the library are randomized, thus requiring a key to work. The MaternFeatureMapKernel uses this key to make them (and thus the kernel) deterministic, applying the utility function make_deterministic() to the pair feature_map, key.

  Note

  Even if the feature_map is deterministic, you need to provide a valid key, although it will essentially be ignored. In the future, we should probably make the key parameter optional.

• normalize (bool) –

  This parameter is directly passed on to the feature_map as a keyword argument “normalize”. If normalize=True, then either \(k(x, x) = 1\) for all \(x \in X\), or \(\int_X k(x, x) d x = 1\), depending on the type of the feature map and on the space \(X\).

  Note

  For many kernel methods, \(k(\cdot, \cdot)\) and \(a k(\cdot, \cdot)\) are indistinguishable, whatever the positive constant \(a\) is. For these, it makes sense to use normalize=False to save up some computational overhead. For others, like for the Gaussian process regression, the normalization of the kernel might be important. In these cases, you will typically want to set normalize=True.

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 (lab.Numeric) – 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 (beartype.typing.Optional[lab.Numeric]) –

  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.

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, **kwargs)

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 (beartype.typing.Dict[str, lab.Numeric]) – Same as for K().

• X (lab.Numeric) – 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()

Initializes the dict of the trainable parameters of the kernel.

Returns dict(nu=np.array([np.inf]), lengthscale=np.array([1.0])).

This dict can be modified and is passed around into such methods as K() or K_diag(), as the params argument.

Note

The values in the returned dict are always of the NumPy array type. Thus, if you want to use some other backend for internal computations when calling K() or K_diag(), you need to replace the values with the analogs typed as arrays of the desired backend.

Return type:

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