code: def test_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_embeddings = (
          torch.eye(2, device=device)
          + (1 - torch.eye(2, device=device)) * 1e-1
      )
      seed_embeddings = seed_embeddings.requires_grad_()
      assert seed_embeddings.grad is None
  
      pca = PCABiasDirection(requires_grad=True)
      test_bias_direction = pca(seed_embeddings)
      test_bias_direction.sum().backward()
      assert seed_embeddings.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 PCABiasDirectionTest obj's underlying member method code with example usage: obj.test_pca_with_grad(device) without return types