code: def covar_cache(self):
      # Get inverse root
      train_train_covar = self.train_prior_dist.lazy_covariance_matrix
      train_interp_indices = train_train_covar.left_interp_indices
      train_interp_values = train_train_covar.left_interp_values
  
      # Get probe vectors for inverse root
      num_probe_vectors = settings.fast_pred_var.num_probe_vectors()
      num_inducing = train_train_covar.base_linear_op.size(-1)
      vector_indices = torch.randperm(num_inducing).type_as(
          train_interp_indices
      )
      probe_vector_indices = vector_indices[:num_probe_vectors]
      test_vector_indices = vector_indices[
          num_probe_vectors : 2 * num_probe_vectors
      ]
  
      probe_interp_indices = probe_vector_indices.unsqueeze(1)
      probe_test_interp_indices = test_vector_indices.unsqueeze(1)
      dtype = train_train_covar.dtype
      device = train_train_covar.device
      probe_interp_values = torch.ones(
          num_probe_vectors, 1, dtype=dtype, device=device
      )
  
      batch_shape = train_train_covar.base_linear_op.batch_shape
      probe_vectors = InterpolatedLinearOperator(
          train_train_covar.base_linear_op,
          train_interp_indices.expand(
              *batch_shape, *train_interp_indices.shape[-2:]
          ),
          train_interp_values.expand(
              *batch_shape, *train_interp_values.shape[-2:]
          ),
          probe_interp_indices.expand(
              *batch_shape, *probe_interp_indices.shape[-2:]
          ),
          probe_interp_values.expand(
              *batch_shape, *probe_interp_values.shape[-2:]
          ),
      ).to_dense()
      test_vectors = InterpolatedLinearOperator(
          train_train_covar.base_linear_op,
          train_interp_indices.expand(
              *batch_shape, *train_interp_indices.shape[-2:]
          ),
          train_interp_values.expand(
              *batch_shape, *train_interp_values.shape[-2:]
          ),
          probe_test_interp_indices.expand(
              *batch_shape, *probe_test_interp_indices.shape[-2:]
          ),
          probe_interp_values.expand(
              *batch_shape, *probe_interp_values.shape[-2:]
          ),
      ).to_dense()
  
      # Put data through the likelihood
      dist = self.train_prior_dist.__class__(
          torch.zeros_like(self.train_prior_dist.mean),
          self.train_prior_dist.lazy_covariance_matrix,
      )
      train_train_covar_plus_noise = self.likelihood(
          dist, self.train_inputs
      ).lazy_covariance_matrix
  
      # Get inverse root
      train_train_covar_inv_root = (
          train_train_covar_plus_noise.root_inv_decomposition(
              initial_vectors=probe_vectors, test_vectors=test_vectors
          ).root
      )
      train_train_covar_inv_root = train_train_covar_inv_root.to_dense()
  
      # New root factor
      root = self._exact_predictive_covar_inv_quad_form_cache(
          train_train_covar_inv_root, self._last_test_train_covar
      )
  
      # Precomputed factor
      if settings.fast_pred_samples.on():
          inside = (
              train_train_covar.base_linear_op
              + RootLinearOperator(root).mul(-1)
          )
          inside_root = inside.root_decomposition().root.to_dense()
          # Prevent backprop through this variable
          if settings.detach_test_caches.on():
              inside_root = inside_root.detach()
          covar_cache = inside_root, None
      else:
          # Prevent backprop through this variable
          if settings.detach_test_caches.on():
              root = root.detach()
          covar_cache = None, root
  
      return covar_cache
  
imports: import functools
  import string
  import warnings
  import torch
  from linear_operator import to_dense, to_linear_operator
  from linear_operator.operators import AddedDiagLinearOperator, BatchRepeatLinearOperator, ConstantMulLinearOperator, DenseLinearOperator, InterpolatedLinearOperator, LinearOperator, LowRankRootAddedDiagLinearOperator, MaskedLinearOperator, MatmulLinearOperator, RootLinearOperator, ZeroLinearOperator
  from linear_operator.utils.cholesky import psd_safe_cholesky
  from linear_operator.utils.interpolation import left_interp, left_t_interp
  from torch import Tensor
  from ..None import settings
  from ..distributions import MultitaskMultivariateNormal
  from ..lazy import LazyEvaluatedKernelTensor
  from ..utils.memoize import add_to_cache, cached, clear_cache_hook, pop_from_cache
label_desc: Use image node_id 11 for calling the InterpolatedPredictionStrategy obj's underlying member method code with example usage: obj.covar_cache() and returns: covar_cache