code: def test_multiple_inputs_parametrization(self):
      # A parametrization with several outputs
      class RankOne(nn.Module):
          def forward(self, x, y):
              # Form a rank-1 matrix from a pair of vectors
              return x.unsqueeze(-1) @ y.unsqueeze(-2)
  
          def right_inverse(self, Y):
              # We project the given matrix onto the rank 1 matrices
              U, S, Vh = torch.linalg.svd(Y, full_matrices=False)
              # S is ordered in a decreasing way.
              s0_sqrt = S[0].sqrt().unsqueeze(-1)
              return U[..., :, 0] * s0_sqrt, Vh[..., 0, :] * s0_sqrt
  
      # Simple parametrisation
      class Double(nn.Module):
          def forward(self, x):
              return 2.0 * x
  
          def right_inverse(self, w):
              return 0.5 * w
  
      model = nn.Linear(3, 3)
      # Test one parametrization
      parametrize.register_parametrization(model, "weight", RankOne())
      self.assertTrue(hasattr(model, "parametrizations"))
      self.assertTrue(parametrize.is_parametrized(model))
      self.assertTrue(parametrize.is_parametrized(model, "weight"))
      self.assertTrue(
          hasattr(model.parametrizations.weight, "original0")
      )
      self.assertIn(
          "original0", model.parametrizations.weight._parameters
      )
      self.assertTrue(
          hasattr(model.parametrizations.weight, "original1")
      )
      self.assertIn(
          "original1", model.parametrizations.weight._parameters
      )
      self.assertFalse(parametrize.is_parametrized(model, "bias"))
      self.assertNotIn("weight", model._parameters)
      # Result should be rank 1
      self.assertEqual(torch.linalg.matrix_rank(model.weight).item(), 1)
  
      with self.assertRaisesRegex(
          ValueError, "leave_parametrized=False"
      ):
          # Cannot remove a parametrization with multiple inputs and not leave it parametrized
          parametrize.remove_parametrizations(
              model, "weight", leave_parametrized=False
          )
      # Remove parametrization and check consistency
      parametrize.remove_parametrizations(
          model, "weight", leave_parametrized=True
      )
      self.assertFalse(hasattr(model, "parametrizations"))
      self.assertEqual(model.__class__, nn.Linear)
      self.assertFalse(parametrize.is_parametrized(model))
      self.assertEqual(torch.linalg.matrix_rank(model.weight).item(), 1)
      self.assertIn("weight", model._parameters)
  
      # Registering parametrizations with one input on top of one with multiple inputs should work
      init_weight = model.weight.clone()
      parametrize.register_parametrization(model, "weight", RankOne())
      # Projecting a rank 1 matrix onto the matrices of rank one does not change the matrix
      self.assertEqual(init_weight, model.weight)
      parametrize.register_parametrization(model, "weight", Double())
      # The matrix now is twice the initial matrix
      self.assertEqual(2.0 * init_weight, model.weight)
      # Multiplying by a scalar does not change the rank
      self.assertEqual(torch.linalg.matrix_rank(model.weight).item(), 1)
  
      # The model has now three parameters
      self.assertEqual(len(list(model.parameters())), 3)
  
      sgd = torch.optim.SGD(model.parameters(), lr=0.1)
  
      # Test backward. Should not throw
      for _ in range(2):
          sgd.zero_grad()
          loss = (model.weight.T @ model.bias).sum()
          loss.backward()
          sgd.step()
  
      # Same drill as before, removing should work as expected
      with self.assertRaisesRegex(
          ValueError, "leave_parametrized=False"
      ):
          # Cannot remove a parametrization with multiple inputs and not leave it parametrized
          parametrize.remove_parametrizations(
              model, "weight", leave_parametrized=False
          )
      # Remove parametrization and check consistency
      parametrize.remove_parametrizations(
          model, "weight", leave_parametrized=True
      )
      self.assertFalse(hasattr(model, "parametrizations"))
      self.assertEqual(model.__class__, nn.Linear)
      self.assertFalse(parametrize.is_parametrized(model))
      self.assertEqual(torch.linalg.matrix_rank(model.weight).item(), 1)
      self.assertIn("weight", model._parameters)
  
      # The model has now two parameters
      self.assertEqual(len(list(model.parameters())), 2)
  
      # Test backward. Should not throw
      sgd = torch.optim.SGD(model.parameters(), lr=0.1)
      for _ in range(2):
          sgd.zero_grad()
          loss = (model.weight.T @ model.bias).sum()
          loss.backward()
          sgd.step()
  
imports: from copy import deepcopy
  from itertools import product
  import pickle
  import torch
  import torch.nn
  import torch.nn.functional
  import torch.nn.init
  import torch.nn.utils.parametrize
  from torch.nn import Parameter
  from torch.testing._internal.common_utils import run_tests, skipIfNoLapack, TemporaryFileName, instantiate_parametrized_tests, set_default_dtype
  from torch.testing._internal.common_device_type import instantiate_device_type_tests
  from torch.testing._internal.common_cuda import TEST_MULTIGPU
  from torch.testing._internal.common_nn import NNTestCase
  from torch.testing._internal.common_utils import gradcheck
label_desc: Use image node_id 8 for calling the TestNNParametrization obj's underlying member method code with example usage: obj.test_multiple_inputs_parametrization() and returns: unknown, unknown,  unknown, unknown, unknown