code: def gauss_kl_vff(q_mu, q_sqrt, K):
      """
      Compute the KL divergence from
  
            q(x) = N(q_mu, q_sqrt^2)
      to
            p(x) = N(0, K)
  
      q_mu is a vector [N, 1] that contains the mean.
      q_sqrt is a matrix that is the lower triangular square-root matrix of the covariance of q.
  
      K is a positive definite matrix: the covariance of p.
      NOTE: K is a LinearOperator that provides efficient methjods
          for solve(), log_abs_determinant(), and trace()
      """
      # KL(N₀ || N₁) = ½ [tr(Σ₁⁻¹ Σ₀) + (μ₁ - μ₀)ᵀ Σ₁⁻¹ (μ₁ - μ₀) - k + ln(det(Σ₁)/det(Σ₀))]
      # N₀ = q; μ₀ = q_mu, Σ₀ = q_sqrt q_sqrtᵀ
      # N₁ = p; μ₁ = 0, Σ₁ = K
      # KL(q || p) =
      #     ½ [tr(K⁻¹ q_sqrt q_sqrtᵀA + q_muᵀ K⁻¹ q_mu - k + logdet(K) - logdet(q_sqrt q_sqrtᵀ)]
      # k = number of dimensions, if q_sqrt is m x m this is m²
      Kinv_q_mu = K.solve(q_mu)
  
      mahalanobis_term = tf.squeeze(
          tf.matmul(q_mu, Kinv_q_mu, transpose_a=True)
      )
  
      # GPflow: q_sqrt is num_latent_gps x N x N
      num_latent_gps = to_default_float(tf.shape(q_mu)[1])
      logdet_prior = num_latent_gps * K.log_abs_determinant()
  
      product_of_dimensions__int = tf.reduce_prod(
          tf.shape(q_sqrt)[:-1]
      )  # dimensions are integers
      constant_term = to_default_float(product_of_dimensions__int)
  
      Lq = tf.linalg.band_part(q_sqrt, -1, 0)  # force lower triangle
      logdet_q = tf.reduce_sum(
          tf.math.log(tf.square(tf.linalg.diag_part(Lq)))
      )
  
      # S = tf.matmul(q_sqrt, q_sqrt, transpose_b=True)
      # trace_term = tf.trace(K.solve(S))
      trace_term = tf.squeeze(
          tf.reduce_sum(Lq * K.solve(Lq), axis=[-1, -2])
      )  # [O(N²) instead of O(N³)
  
      twoKL = (
          trace_term
          + mahalanobis_term
          - constant_term
          + logdet_prior
          - logdet_q
      )
      return 0.5 * twoKL
  
imports: import numpy
  import tensorflow
  from check_shapes import Shape
  import gpflow
  from gpflow import covariances
  from gpflow import kullback_leiblers
  from gpflow.base import TensorLike
  from gpflow.ci_utils import reduce_in_tests
  from gpflow.inducing_variables import InducingVariables
  from gpflow.utilities import to_default_float
  import matplotlib.pyplot
  import gpflow.posteriors
label_desc: Use image node_id 6 for calling a global function with example usage: gauss_kl_vff(q_mu, q_sqrt, K) and returns: unknown