code: def test_cholesky_transform(batch_shape, dim, transform):
      arange = torch.arange(dim)
      domain = transform.domain
      z = torch.randn(batch_shape + (dim * (dim - 1) // 2,))
      if domain is dist.constraints.corr_matrix:
          tril_mask = arange < arange.view(-1, 1)
      else:
          tril_mask = arange < arange.view(-1, 1) + 1
      x = transform.inv(
          T.CorrCholeskyTransform()(z)
      )  # creates corr_matrix
  
      def vec_to_mat(x_vec):
          x_mat = x_vec.new_zeros(batch_shape + (dim, dim))
          x_mat[..., tril_mask] = x_vec
          x_mat = (
              x_mat
              + x_mat.transpose(-2, -1)
              - x_mat.diagonal(dim1=-2, dim2=-1).diag_embed()
          )
          if domain == dist.constraints.corr_matrix:
              x_mat = (
                  x_mat + x_mat.new_ones(x_mat.shape[-1]).diag_embed()
              )
          return x_mat
  
      def transform_to_vec(x_vec):
          x_mat = vec_to_mat(x_vec)
          return transform(x_mat)[..., tril_mask]
  
      x_vec = x[..., tril_mask].clone().requires_grad_()
      x_mat = vec_to_mat(x_vec)
      y = transform(x_mat)
      log_det = transform.log_abs_det_jacobian(x_mat, y)
      if batch_shape == ():
          jacobian = torch.autograd.functional.jacobian(
              transform_to_vec, x_vec
          )
          assert_close(log_det, torch.slogdet(jacobian)[1])
  
      assert log_det.shape == batch_shape
      assert_close(y, torch.linalg.cholesky(x_mat))
      assert_close(transform.inv(y), x_mat)
  
imports: import operator
  from functools import partial, reduce
  from unittest import TestCase
  import pytest
  import torch
  import pyro.distributions
  import pyro.distributions.transforms
  from pyro.distributions import constraints
  from tests.common import assert_close
label_desc: Use image node_id 1 for calling a global function with example usage: test_cholesky_transform(batch_shape, dim, transform) and returns: x_mat, transform