code: def _gaussian_update_ApproximateGP_VariationalStrategy(
      model: ApproximateGP,
      _: VariationalStrategy,
      *,
      sample_values: Tensor,
      target_values: Optional[Tensor] = None,
      noise_covariance: Optional[Union[Tensor, LinearOperator]] = None,
      input_transform: Optional[InputTransform] = None,
      **ignore: Any,
  ) -> GeneralizedLinearPath:
      # TODO: Account for jitter added by `psd_safe_cholesky`
      if not isinstance(
          noise_covariance, (NoneType, ZeroLinearOperator)
      ):
          raise NotImplementedError(
              f"`noise_covariance` argument not yet supported for {type(model)}."
          )
  
      # Inducing points `Z` are assumed to live in transformed space
      batch_shape = model.covar_module.batch_shape
      v = model.variational_strategy
      Z = v.inducing_points
      L = v._cholesky_factor(
          v(Z, prior=True).lazy_covariance_matrix
      ).to(dtype=sample_values.dtype)
  
      # Generate whitened inducing variables `u`, then location-scale transform
      if target_values is None:
          u = v.variational_distribution.rsample(
              sample_values.shape[
                  : sample_values.ndim - len(batch_shape) - 1
              ],
          )
          target_values = model.mean_module(Z) + (
              u @ L.transpose(-1, -2)
          )
  
      return _gaussian_update_exact(
          kernel=model.covar_module,
          points=Z,
          target_values=target_values,
          sample_values=sample_values,
          scale_tril=L,
          input_transform=input_transform,
      )
  
imports: from __future__ import annotations
  from typing import Any, Callable, Optional, Union
  import torch
  from botorch.models.approximate_gp import ApproximateGPyTorchModel
  from botorch.models.transforms.input import InputTransform
  from botorch.sampling.pathwise.features import KernelEvaluationMap
  from botorch.sampling.pathwise.paths import GeneralizedLinearPath, SamplePath
  from botorch.sampling.pathwise.utils import get_input_transform, get_train_inputs, get_train_targets, TInputTransform
  from botorch.utils.dispatcher import Dispatcher
  from botorch.utils.types import DEFAULT, NoneType
  from gpytorch.kernels.kernel import Kernel
  from gpytorch.likelihoods import _GaussianLikelihoodBase, Likelihood
  from gpytorch.models import ApproximateGP, ExactGP, GP
  from gpytorch.variational import VariationalStrategy
  from linear_operator.operators import LinearOperator, SumLinearOperator, ZeroLinearOperator
  from torch import Tensor
label_desc: Use image node_id 6 for calling a global function with example usage: _gaussian_update_ApproximateGP_VariationalStrategy(model, _) and returns: _gaussian_update_exact