from __future__ import absolute_import, division, print_function, unicode_literals
from typing import Optional
import torch
from ..constraints import Interval, Positive
from .kernel import Kernel
[docs]class RQKernel(Kernel):
r"""
Computes a covariance matrix based on the rational quadratic kernel
between inputs :math:`\mathbf{x_1}` and :math:`\mathbf{x_2}`:
.. math::
\begin{equation*}
k_{\text{RQ}}(\mathbf{x_1}, \mathbf{x_2}) = \left(1 + \frac{1}{2\alpha}
(\mathbf{x_1} - \mathbf{x_2})^\top \Theta^{-2} (\mathbf{x_1} - \mathbf{x_2}) \right)^{-\alpha}
\end{equation*}
where :math:`\Theta` is a lengthscale parameter, and :math:`\alpha` is the
rational quadratic relative weighting parameter.
See :class:`gpytorch.kernels.Kernel` for descriptions of the lengthscale options.
.. note::
This kernel does not have an `outputscale` parameter. To add a scaling parameter,
decorate this kernel with a :class:`gpytorch.kernels.ScaleKernel`.
Args:
ard_num_dims (int, optional):
Set this if you want a separate lengthscale for each
input dimension. It should be `d` if x1 is a `n x d` matrix. Default: `None`
batch_shape (torch.Size, optional):
Set this if you want a separate lengthscale for each
batch of input data. It should be `b` if x1 is a `b x n x d` tensor. Default: `torch.Size([])`.
active_dims (tuple of ints, optional):
Set this if you want to compute the covariance of only a few input dimensions. The ints
corresponds to the indices of the dimensions. Default: `None`.
lengthscale_prior (Prior, optional):
Set this if you want to apply a prior to the lengthscale parameter. Default: `None`.
lengthscale_constraint (Constraint, optional):
Set this if you want to apply a constraint to the lengthscale parameter. Default: `Positive`.
alpha_constraint (Constraint, optional):
Set this if you want to apply a constraint to the alpha parameter. Default: `Positive`.
eps (float):
The minimum value that the lengthscale can take (prevents divide by zero errors). Default: `1e-6`.
Attributes:
lengthscale (Tensor):
The lengthscale parameter. Size/shape of parameter depends on the
ard_num_dims and batch_shape arguments.
alpha (Tensor):
The rational quadratic relative weighting parameter. Size/shape of parameter depends
on the batch_shape argument
"""
has_lengthscale = True
def __init__(self, alpha_constraint: Optional[Interval] = None, **kwargs):
super(RQKernel, self).__init__(**kwargs)
self.register_parameter(name="raw_alpha", parameter=torch.nn.Parameter(torch.zeros(*self.batch_shape, 1)))
if alpha_constraint is None:
alpha_constraint = Positive()
self.register_constraint("raw_alpha", alpha_constraint)
def forward(self, x1, x2, diag=False, **params):
def postprocess_rq(dist):
alpha = self.alpha
for _ in range(1, len(dist.shape) - len(self.batch_shape)):
alpha = alpha.unsqueeze(-1)
return (1 + dist.div(2 * alpha)).pow(-alpha)
x1_ = x1.div(self.lengthscale)
x2_ = x2.div(self.lengthscale)
return self.covar_dist(
x1_, x2_, square_dist=True, diag=diag, dist_postprocess_func=postprocess_rq, postprocess=True, **params
)
@property
def alpha(self):
return self.raw_alpha_constraint.transform(self.raw_alpha)
@alpha.setter
def alpha(self, value):
if not torch.is_tensor(value):
value = torch.as_tensor(value).to(self.raw_lengthscale)
self.initialize(raw_alpha=self.raw_alpha_constraint.inverse_transform(value))