code: def test_paired_pca_with_grad(self, device: str):
      # add noise to avoid "RuntimeError: triangular_solve_cpu: U(2,2) is zero, singular U."
      torch.manual_seed(0)
      seed_embeddings1 = torch.tensor(
          [[1.0, 1.0], [1.0, 1.0]], device=device
      )
      seed_embeddings2 = (1 - torch.eye(2, device=device)) * 9e-1
      seed_embeddings1 = seed_embeddings1.requires_grad_()
      seed_embeddings2 = seed_embeddings2.requires_grad_()
      assert seed_embeddings1.grad is None
      assert seed_embeddings2.grad is None
  
      paired_pca = PairedPCABiasDirection(requires_grad=True)
      test_bias_direction = paired_pca(
          seed_embeddings1, seed_embeddings2
      )
      test_bias_direction.sum().backward()
      assert seed_embeddings1.grad is not None
      assert seed_embeddings2.grad is not None
  
imports: import torch
  from torch import allclose
  import pytest
  import math
  from allennlp.common.checks import ConfigurationError
  from allennlp.common.testing import AllenNlpTestCase, multi_device
  from allennlp.fairness.bias_direction import PCABiasDirection, PairedPCABiasDirection, TwoMeansBiasDirection, ClassificationNormalBiasDirection
label_desc: Use image node_id 3 for calling the PairedPCABiasDirectionTest obj's underlying member method code with example usage: obj.test_paired_pca_with_grad(device) without return types