code: def update_inverse(a_inv: types.Float, x: types.Float) -> types.Float:
      """Updates the inverse using the Woodbury matrix identity.
  
      Given a matrix `A` of size d-by-d and a matrix `X` of size k-by-d, this
      function computes the inverse of B = A + X^T X, assuming that the inverse of
      A is available.
  
      Reference:
      https://en.wikipedia.org/wiki/Woodbury_matrix_identity
  
      Args:
        a_inv: a `Tensor` of shape [`d`, `d`]. This is the current inverse of `A`.
        x: a `Tensor` of shape [`k`, `d`].
  
      Returns:
        The update that needs to be added to 'a_inv' to compute the inverse.
        If `x` is empty, a zero matrix is returned.
      """
      batch_size = tf.shape(x)[0]
  
      def true_fn():
          return tf.zeros_like(a_inv)
  
      def false_fn():
          return simplified_woodbury_update(a_inv, x)
  
      a_inv_update = tf.cond(tf.equal(batch_size, 0), true_fn, false_fn)
      return a_inv_update
  
file_path: agents/tf_agents/bandits/policies/linalg.py
imports: from __future__ import absolute_import
  from __future__ import division
  from __future__ import print_function
  import tensorflow
  from tf_agents.typing import types
  from tf_agents.utils import common
label_desc: Use image node_id 5 for calling a global function with example usage: update_inverse(a_inv, x) and returns: a_inv_update, tf, simplified_woodbury_update