code: def covar_cache(self):
      # Here, the covar_cache is going to be K_{UU}^{-1/2} K_{UX}( K_{XX} + \sigma^2 I )^{-1} K_{XU} K_{UU}^{-1/2}
      # This is easily computed using Woodbury
      # K_{XX} + \sigma^2 I = R R^T + \sigma^2 I
      #                     = \sigma^{-2} ( I - \sigma^{-2} R (I + \sigma^{-2} R^T R)^{-1} R^T  )
      train_train_covar = self.lik_train_train_covar.evaluate_kernel()
  
      # Get terms needed for woodbury
      root = (
          train_train_covar._linear_op.root_decomposition().root.to_dense()
      )  # R
      inv_diag = train_train_covar._diag_tensor.inverse()  # \sigma^{-2}
  
      # Form LT using woodbury
      ones = torch.tensor(1.0, dtype=root.dtype, device=root.device)
      chol_factor = to_linear_operator(
          root.transpose(-1, -2) @ (inv_diag @ root)
      ).add_diagonal(
          ones
      )  # (I + \sigma^{-2} R^T R)^{-1}
      woodbury_term = inv_diag @ torch.linalg.solve_triangular(
          chol_factor.cholesky().to_dense(),
          root.transpose(-1, -2),
          upper=False,
      ).transpose(-1, -2)
      # woodbury_term @ woodbury_term^T = \sigma^{-2} R (I + \sigma^{-2} R^T R)^{-1} R^T \sigma^{-2}
  
      inverse = AddedDiagLinearOperator(
          inv_diag,
          MatmulLinearOperator(
              -woodbury_term, woodbury_term.transpose(-1, -2)
          ),
      )
      # \sigma^{-2} ( I - \sigma^{-2} R (I + \sigma^{-2} R^T R)^{-1} R^T  )
  
      return root.transpose(-1, -2) @ (inverse @ root)
  
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 1 for calling the SGPRPredictionStrategy obj's underlying member method code with example usage: obj.covar_cache() and returns: unknown