#!/usr/bin/env python3
import math
import torch
from linear_operator.operators import KroneckerProductLinearOperator
from gpytorch.kernels.matern_kernel import MaternKernel
sqrt5 = math.sqrt(5)
five_thirds = 5.0 / 3.0
[docs]class Matern52KernelGradGrad(MaternKernel):
r"""
Computes a covariance matrix of the Matern52 kernel that models the covariance
between the values and first and second (non-mixed) partial derivatives for inputs :math:`\mathbf{x_1}`
and :math:`\mathbf{x_2}`.
See :class:`qpytorch.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 = qpytorch.kernels.ScaleKernel(qpytorch.kernels.Matern52KernelGradGrad())
>>> covar = covar_module(x) # Output: LinearOperator of size (110 x 110), where 110 = n * (2*d + 1)
>>>
>>> batch_x = torch.randn(2, 10, 5)
>>> # Batch: Simple option
>>> covar_module = qpytorch.kernels.ScaleKernel(qpytorch.kernels.Matern52KernelGradGrad())
>>> # Batch: different lengthscale for each batch
>>> covar_module = qpytorch.kernels.ScaleKernel(qpytorch.kernels.Matern52KernelGradGrad(batch_shape=torch.Size([2]))) # noqa: E501
>>> covar = covar_module(x) # Output: LinearOperator of size (2 x 110 x 110)
"""
def __init__(self, **kwargs):
# remove nu in case it was set
kwargs.pop("nu", None)
super(Matern52KernelGradGrad, self).__init__(nu=2.5, **kwargs)
self._interleaved = kwargs.pop('interleaved', True)
def forward(self, x1, x2, diag=False, **params):
lengthscale = self.lengthscale
batch_shape = x1.shape[:-2]
n_batch_dims = len(batch_shape)
n1, d = x1.shape[-2:]
n2 = x2.shape[-2]
mask_idx1 = params.pop('mask_idx1', None) # mask off-diagonal covariance
if mask_idx1 is not None:
mask_idx2 = params.pop('mask_idx2', mask_idx1)
assert mask_idx1.shape[:-1] == mask_idx2.shape[:-1], 'Batch shapes of mask indices do not match!'
if not diag:
K = torch.zeros(*batch_shape, n1 * (2 * d + 1), n2 * (2 * d + 1), device=x1.device, dtype=x1.dtype)
distance_matrix = self.covar_dist(x1.div(lengthscale), x2.div(lengthscale), diag=diag, **params)
exp_neg_sqrt5r = torch.exp(-sqrt5 * distance_matrix)
one_plus_sqrt5r = 1 + sqrt5 * distance_matrix
# differences matrix in each dimension to be used for derivatives
# shape of n1 x n2 x d
outer = x1.view(*batch_shape, n1, 1, d) - x2.view(*batch_shape, 1, n2, d)
outer = outer / lengthscale.unsqueeze(-2) ** 2
# shape of n1 x d x n2
outer = torch.transpose(outer, -1, -2).contiguous()
# 1) Kernel block, cov(f^m, f^n)
# shape is n1 x n2
# exp_component = torch.exp(-sqrt5 * distance_matrix)
constant_component = one_plus_sqrt5r.add(five_thirds * distance_matrix**2)
K[..., :n1, :n2] = constant_component * exp_neg_sqrt5r #exp_component
# 2) First gradient block, cov(f^m, omega^n_i)
outer1 = outer.view(*batch_shape, n1, n2 * d)
# the - signs on -outer1 and -five_thirds cancel out
K[..., :n1, n2: (n2 * (d + 1))] = five_thirds * outer1 * (one_plus_sqrt5r * exp_neg_sqrt5r).repeat(
[*([1] * (n_batch_dims + 1)), d]
)
# 3) Second gradient block, cov(omega^m_j, f^n)
outer2 = outer.transpose(-1, -3).reshape(*batch_shape, n2, n1 * d)
outer2 = outer2.transpose(-1, -2)
K[..., n1: (n1 * (d + 1)), :n2] = -five_thirds * outer2 * (one_plus_sqrt5r * exp_neg_sqrt5r).repeat(
[*([1] * n_batch_dims), d, 1]
)
# 4) Hessian block, cov(omega^m_j, omega^n_i)
outer3 = outer1.repeat([*([1] * n_batch_dims), d, 1]) * outer2.repeat([*([1] * (n_batch_dims + 1)), d])
kp = KroneckerProductLinearOperator(
torch.eye(d, d, device=x1.device, dtype=x1.dtype).repeat(*batch_shape, 1, 1) / lengthscale**2,
torch.ones(n1, n2, device=x1.device, dtype=x1.dtype).repeat(*batch_shape, 1, 1),
)
# part1 = -five_thirds * exp_neg_sqrt5r
# part2 = 5 * outer3
# part3 = 1 + sqrt5 * distance_matrix
exp_neg_sqrt5rdd = exp_neg_sqrt5r.repeat([*([1] * (n_batch_dims)), d, d])
K[..., n1: (n1 * (d + 1)), n2: (n2 * (d + 1))] = -five_thirds * exp_neg_sqrt5rdd.mul(
# need to use kp.to_dense().mul instead of kp.to_dense().mul_
# because otherwise a RuntimeError is raised due to how autograd works with
# view + inplace operations in the case of 1-dimensional input
(5 * outer3).sub(kp.to_dense().mul(one_plus_sqrt5r.repeat([*([1] * n_batch_dims), d, d])))
)
# 5) 1-3 block
douter1dx2 = KroneckerProductLinearOperator(
torch.ones(1, d, device=x1.device, dtype=x1.dtype).repeat(*batch_shape, 1, 1) / self.lengthscale.pow(2),
torch.ones(n1, n2, device=x1.device, dtype=x1.dtype).repeat(*batch_shape, 1, 1),
).to_dense()
K_13 = five_thirds * (-douter1dx2 * one_plus_sqrt5r.repeat([*([1] * (n_batch_dims + 1)), d]) + 5* outer1 * outer1) * exp_neg_sqrt5r.repeat(
[*([1] * (n_batch_dims + 1)), d]
) # verified for n1=n2=1 case
K[..., :n1, (n2 * (d + 1)) :] = K_13
if d>1:
douter1dx2 = KroneckerProductLinearOperator(
(torch.ones(1, d, device=x1.device, dtype=x1.dtype).repeat(*batch_shape, 1, 1) / self.lengthscale.pow(2)).transpose(-1, -2),
torch.ones(n1, n2, device=x1.device, dtype=x1.dtype).repeat(*batch_shape, 1, 1),
).to_dense()
K_31 = five_thirds * (-douter1dx2 * one_plus_sqrt5r.repeat([*([1] * n_batch_dims), d, 1]) + 5* outer2 * outer2) * exp_neg_sqrt5r.repeat(
[*([1] * n_batch_dims), d, 1]
) # verified for n1=n2=1 case
K[..., (n1 * (d + 1)) :, :n2] = K_31
# rest of the blocks are all of size (n1*d,n2*d)
outer1 = outer1.repeat([*([1] * n_batch_dims), d, 1])
outer2 = outer2.repeat([*([1] * (n_batch_dims + 1)), d])
# II = (torch.eye(d,d,device=x1.device,dtype=x1.dtype)/lengthscale.pow(2)).repeat(*batch_shape,n1,n2)
kp2 = KroneckerProductLinearOperator(
torch.ones(d, d, device=x1.device, dtype=x1.dtype).repeat(*batch_shape, 1, 1) / self.lengthscale.pow(2),
torch.ones(n1, n2, device=x1.device, dtype=x1.dtype).repeat(*batch_shape, 1, 1),
).to_dense()
# II may not be the correct thing to use. It might be more appropriate to use kp instead??
II = kp.to_dense()
# exp_neg_sqrt5rdd = exp_neg_sqrt5r.repeat([*([1] * (n_batch_dims)), d, d])
invrdd = (distance_matrix+self.eps).pow(-1)
# invrdd[torch.arange(min(n1,n2)),torch.arange(min(n1,n2))] = distance_matrix.diagonal()
invrdd = invrdd.repeat([*([1] * (n_batch_dims)), d, d])
# invrdd = distance_matrix.pow(-1).fill_diagonal_(0).repeat([*([1] * (n_batch_dims)), d, d]).fill_diagonal_(1)
K_23 = five_thirds * 5* ((kp2 - sqrt5*invrdd* outer1 * outer1) * outer2 + 2.0 * II * outer1) * exp_neg_sqrt5rdd # verified for n1=n2=1 case
K[..., n1 : (n1 * (d + 1)), (n2 * (d + 1)) :] = K_23
if d>1:
kp2t = KroneckerProductLinearOperator(
(torch.ones(d, d, device=x1.device, dtype=x1.dtype).repeat(*batch_shape, 1, 1) / self.lengthscale.pow(2)).transpose(-1, -2),
torch.ones(n1, n2, device=x1.device, dtype=x1.dtype).repeat(*batch_shape, 1, 1),
).to_dense()
K_32 = five_thirds * 5* (
(-(kp2t if d>1 else kp2) + sqrt5*invrdd* outer2 * outer2) * outer1 - 2.0 * II * outer2
) * exp_neg_sqrt5rdd # verified for n1=n2=1 case
K[..., (n1 * (d + 1)) :, n2 : (n2 * (d + 1))] = K_32
# K_33 = five_thirds * 5*(
# ((-(kp2t if d>1 else kp2) + sqrt5*invrdd*outer2 * outer2) * (-kp2) - 2.0 *sqrt5*invrdd * II * outer2 * outer1 + 2.0 * (II) ** 2
# ) + (
# (-(kp2t if d>1 else kp2)*sqrt5*invrdd + (5+sqrt5*invrdd)*invrdd**2*outer2 * outer2) * outer1 - 2.0 *sqrt5*invrdd * II * outer2
# ) * outer1) * exp_neg_sqrt5rdd # verified for n1=n2=1 case
K_33 = five_thirds * 5*(
(kp2 - sqrt5*invrdd*outer1 * outer1) * ((kp2t if d>1 else kp2)-sqrt5*invrdd* outer2 * outer2)
+ sqrt5*invrdd*(invrdd**2*outer3-4*(II))*outer3 + 2*(II)**2
) * exp_neg_sqrt5rdd
K[..., (n1 * (d + 1)) :, (n2 * (d + 1)) :] = K_33
# Symmetrize for stability
if n1 == n2 and torch.eq(x1, x2).all():
K = 0.5 * (K.transpose(-1, -2) + K)
# Apply a perfect shuffle permutation to match the MutiTask ordering
if self._interleaved:
pi1 = torch.arange(n1 * (2 * d + 1)).view(2 * d + 1, n1).t().reshape((n1 * (2 * d + 1)))
pi2 = torch.arange(n2 * (2 * d + 1)).view(2 * d + 1, n2).t().reshape((n2 * (2 * d + 1)))
K = K[..., pi1, :][..., :, pi2]
if mask_idx1 is not None:
diag2keep = K.diagonal(dim1=-1, dim2=-2)[mask_idx1&mask_idx2]
keep_idx = (~mask_idx1).unsqueeze(-1) & (~mask_idx2).unsqueeze(-2)
K = K * keep_idx
K.diagonal(dim1=-1, dim2=-2)[mask_idx1&mask_idx2] = diag2keep * self.eps
# if mask_idx1.ndim==1:
# diag2keep = K[...,mask_idx1,mask_idx2]
# K[...,mask_idx1,:] = 0; K[...,mask_idx2] = 0
# K[...,mask_idx1,mask_idx2] = diag2keep * self.eps
# elif mask_idx1.ndim==2:
# for b in range(mask_idx1.shape[0]):
# diag2keep = K[b,...,mask_idx1[b],mask_idx2[b]]
# K[b,...,mask_idx1[b],:] = 0; K[b,...,mask_idx2[b]] = 0
# K[b,...,mask_idx1[b],mask_idx2[b]] = diag2keep * self.eps
# else:
# raise NotImplementedError('Mask indices of batch dimension bigger than 1 not implemented!')
return K
else:
if not (n1 == n2 and torch.eq(x1, x2).all()):
raise RuntimeError("diag=True only works when x1 == x2")
# nu is set to 2.5
kernel_diag = super(Matern52KernelGradGrad, self).forward(x1, x2, diag=True)
grad_diag = (
five_thirds * torch.ones(*batch_shape, n2, d, device=x1.device, dtype=x1.dtype)
) / lengthscale**2
grad_diag = grad_diag.transpose(-1, -2).reshape(*batch_shape, n2 * d)
gradgrad_diag = (
5**2 * torch.ones(*batch_shape, n2, d, device=x1.device, dtype=x1.dtype) / lengthscale.pow(4)
)
gradgrad_diag = gradgrad_diag.transpose(-1, -2).reshape(*batch_shape, n2 * d)
k_diag = torch.cat((kernel_diag, grad_diag, gradgrad_diag), dim=-1)
if self._interleaved:
pi = torch.arange(n2 * (2 * d + 1)).view(2 * d + 1, n2).t().reshape((n2 * (2 * d + 1)))
k_diag = k_diag[..., pi]
return k_diag
def num_outputs_per_input(self, x1, x2):
return x1.size(-1) * 2 + 1