#!/usr/bin/env python3
from ..functions import RBFCovariance
from ..settings import trace_mode
from .kernel import Kernel
def postprocess_rbf(dist_mat):
return dist_mat.div_(-2).exp_()
[docs]class RBFKernel(Kernel):
r"""
Computes a covariance matrix based on the RBF (squared exponential) kernel
between inputs :math:`\mathbf{x_1}` and :math:`\mathbf{x_2}`:
.. math::
\begin{equation*}
k_{\text{RBF}}(\mathbf{x_1}, \mathbf{x_2}) = \exp \left( -\frac{1}{2}
(\mathbf{x_1} - \mathbf{x_2})^\top \Theta^{-2} (\mathbf{x_1} - \mathbf{x_2}) \right)
\end{equation*}
where :math:`\Theta` is a lengthscale 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`.
:param ard_num_dims: 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`.)
:param batch_shape: Set this if you want a separate lengthscale for each batch of input
data. It should be :math:`B_1 \times \ldots \times B_k` if :math:`\mathbf x1` is
a :math:`B_1 \times \ldots \times B_k \times N \times D` tensor.
:param active_dims: 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`.)
:param lengthscale_prior: Set this if you want to apply a prior to the
lengthscale parameter. (Default: `None`)
:param lengthscale_constraint: Set this if you want to apply a constraint
to the lengthscale parameter. (Default: `Positive`.)
:param eps: The minimum value that the lengthscale can take (prevents
divide by zero errors). (Default: `1e-6`.)
:ivar torch.Tensor lengthscale: The lengthscale parameter. Size/shape of parameter depends on the
ard_num_dims and batch_shape arguments.
Example:
>>> x = torch.randn(10, 5)
>>> # Non-batch: Simple option
>>> covar_module = gpytorch.kernels.ScaleKernel(gpytorch.kernels.RBFKernel())
>>> # Non-batch: ARD (different lengthscale for each input dimension)
>>> covar_module = gpytorch.kernels.ScaleKernel(gpytorch.kernels.RBFKernel(ard_num_dims=5))
>>> covar = covar_module(x) # Output: LinearOperator of size (10 x 10)
>>>
>>> batch_x = torch.randn(2, 10, 5)
>>> # Batch: Simple option
>>> covar_module = gpytorch.kernels.ScaleKernel(gpytorch.kernels.RBFKernel())
>>> # Batch: different lengthscale for each batch
>>> covar_module = gpytorch.kernels.ScaleKernel(gpytorch.kernels.RBFKernel(batch_shape=torch.Size([2])))
>>> covar = covar_module(x) # Output: LinearOperator of size (2 x 10 x 10)
"""
has_lengthscale = True
def forward(self, x1, x2, diag=False, **params):
if (
x1.requires_grad
or x2.requires_grad
or (self.ard_num_dims is not None and self.ard_num_dims > 1)
or diag
or params.get("last_dim_is_batch", False)
or trace_mode.on()
):
x1_ = x1.div(self.lengthscale)
x2_ = x2.div(self.lengthscale)
return postprocess_rbf(self.covar_dist(x1_, x2_, square_dist=True, diag=diag, **params))
return RBFCovariance.apply(
x1,
x2,
self.lengthscale,
lambda x1, x2: self.covar_dist(x1, x2, square_dist=True, diag=False, **params),
)