Source code for gpytorch.mlls.inducing_point_kernel_added_loss_term

#!/usr/bin/env python3

import torch

from ..distributions import MultivariateNormal
from ..likelihoods import GaussianLikelihood, MultitaskGaussianLikelihood
from .added_loss_term import AddedLossTerm

[docs]class InducingPointKernelAddedLossTerm(AddedLossTerm): r""" An added loss term that computes the additional "regularization trace term" of the SGPR objective function. .. math:: -\frac{1}{2 \sigma^2} \text{Tr} \left( \mathbf K_{\mathbf X \mathbf X} - \mathbf Q \right) where :math:`\mathbf Q = \mathbf K_{\mathbf X \mathbf Z} \mathbf K_{\mathbf Z \mathbf Z}^{-1} \mathbf K_{\mathbf Z \mathbf X}` is the Nystrom approximation of :math:`\mathbf K_{\mathbf X \mathbf X}` given by inducing points :math:`\mathbf Z`, and :math:`\sigma^2` is the observational noise of the Gaussian likelihood. See `Titsias, 2009`_, Eq. 9 for more more information. :param prior_dist: A multivariate normal :math:`\mathcal N ( \mathbf 0, \mathbf K_{\mathbf X \mathbf X} )` with covariance matrix :math:`\mathbf K_{\mathbf X \mathbf X}`. :param variational_dist: A multivariate normal :math:`\mathcal N ( \mathbf 0, \mathbf Q` with covariance matrix :math:`\mathbf Q = \mathbf K_{\mathbf X \mathbf Z} \mathbf K_{\mathbf Z \mathbf Z}^{-1} \mathbf K_{\mathbf Z \mathbf X}`. :param likelihood: The Gaussian likelihood with observational noise :math:`\sigma^2`. .. _Titsias, 2009: """ def __init__( self, prior_dist: MultivariateNormal, variational_dist: MultivariateNormal, likelihood: GaussianLikelihood ): self.prior_dist = prior_dist self.variational_dist = variational_dist self.likelihood = likelihood def loss(self, *params) -> torch.Tensor: prior_covar = self.prior_dist.lazy_covariance_matrix variational_covar = self.variational_dist.lazy_covariance_matrix diag = prior_covar.diagonal(dim1=-1, dim2=-2) - variational_covar.diagonal(dim1=-1, dim2=-2) shape = prior_covar.shape[:-1] if isinstance(self.likelihood, MultitaskGaussianLikelihood): shape = torch.Size([*shape, 1]) diag = diag.unsqueeze(-1) noise_diag = self.likelihood._shaped_noise_covar(shape, *params).diagonal(dim1=-1, dim2=-2) if isinstance(self.likelihood, MultitaskGaussianLikelihood): noise_diag = noise_diag.reshape(*shape[:-1], -1) return -0.5 * (diag / noise_diag).sum(dim=[-1, -2]) else: return -0.5 * (diag / noise_diag).sum(dim=-1)