code: def infidelity(
      forward_func: Callable,
      perturb_func: Callable,
      inputs: TensorOrTupleOfTensorsGeneric,
      attributions: TensorOrTupleOfTensorsGeneric,
      baselines: BaselineType = None,
      additional_forward_args: Any = None,
      target: TargetType = None,
      n_perturb_samples: int = 10,
      max_examples_per_batch: int = None,
      normalize: bool = False,
  ) -> Tensor:
      r"""
      Explanation infidelity represents the expected mean-squared error
      between the explanation multiplied by a meaningful input perturbation
      and the differences between the predictor function at its input
      and perturbed input.
      More details about the measure can be found in the following paper:
      https://arxiv.org/abs/1901.09392
  
      It is derived from the completeness property of well-known attribution
      algorithms and is a computationally more efficient and generalized
      notion of Sensitivy-n. The latter measures correlations between the sum
      of the attributions and the differences of the predictor function at
      its input and fixed baseline. More details about the Sensitivity-n can
      be found here:
      https://arxiv.org/abs/1711.06104
  
      The users can perturb the inputs any desired way by providing any
      perturbation function that takes the inputs (and optionally baselines)
      and returns perturbed inputs or perturbed inputs and corresponding
      perturbations.
  
      This specific implementation is primarily tested for attribution-based
      explanation methods but the idea can be expanded to use for non
      attribution-based interpretability methods as well.
  
      Args:
  
          forward_func (Callable):
                  The forward function of the model or any modification of it.
  
          perturb_func (Callable):
                  The perturbation function of model inputs. This function takes
                  model inputs and optionally baselines as input arguments and returns
                  either a tuple of perturbations and perturbed inputs or just
                  perturbed inputs. For example:
  
                  >>> def my_perturb_func(inputs):
                  >>>   <MY-LOGIC-HERE>
                  >>>   return perturbations, perturbed_inputs
  
                  If we want to only return perturbed inputs and compute
                  perturbations internally then we can wrap perturb_func with
                  `infidelity_perturb_func_decorator` decorator such as:
  
                  >>> from captum.metrics import infidelity_perturb_func_decorator
  
                  >>> @infidelity_perturb_func_decorator(<multipy_by_inputs flag>)
                  >>> def my_perturb_func(inputs):
                  >>>   <MY-LOGIC-HERE>
                  >>>   return perturbed_inputs
  
                  In case `multipy_by_inputs` is False we compute perturbations by
                  `input - perturbed_input` difference and in case `multipy_by_inputs`
                  flag is True we compute it by dividing
                  (input - perturbed_input) by (input - baselines).
                  The user needs to only return perturbed inputs in `perturb_func`
                  as described above.
  
                  `infidelity_perturb_func_decorator` needs to be used with
                  `multipy_by_inputs` flag set to False in case infidelity
                  score is being computed for attribution maps that are local aka
                  that do not factor in inputs in the final attribution score.
                  Such attribution algorithms include Saliency, GradCam, Guided Backprop,
                  or Integrated Gradients and DeepLift attribution scores that are already
                  computed with `multipy_by_inputs=False` flag.
  
                  If there are more than one inputs passed to infidelity function those
                  will be passed to `perturb_func` as tuples in the same order as they
                  are passed to infidelity function.
  
                  If inputs
                   - is a single tensor, the function needs to return a tuple
                     of perturbations and perturbed input such as:
                     perturb, perturbed_input and only perturbed_input in case
                     `infidelity_perturb_func_decorator` is used.
                   - is a tuple of tensors, corresponding perturbations and perturbed
                     inputs must be computed and returned as tuples in the
                     following format:
  
                     (perturb1, perturb2, ... perturbN), (perturbed_input1,
                     perturbed_input2, ... perturbed_inputN)
  
                     Similar to previous case here as well we need to return only
                     perturbed inputs in case `infidelity_perturb_func_decorator`
                     decorates out `perturb_func`.
  
                  It is important to note that for performance reasons `perturb_func`
                  isn't called for each example individually but on a batch of
                  input examples that are repeated `max_examples_per_batch / batch_size`
                  times within the batch.
  
          inputs (Tensor or tuple[Tensor, ...]): Input for which
                  attributions are computed. If forward_func takes a single
                  tensor as input, a single input tensor should be provided.
                  If forward_func takes multiple tensors as input, a tuple
                  of the input tensors should be provided. It is assumed
                  that for all given input tensors, dimension 0 corresponds
                  to the number of examples (aka batch size), and if
                  multiple input tensors are provided, the examples must
                  be aligned appropriately.
  
          baselines (scalar, Tensor, tuple of scalar, or Tensor, optional):
                  Baselines define reference values which sometimes represent ablated
                  values and are used to compare with the actual inputs to compute
                  importance scores in attribution algorithms. They can be represented
                  as:
  
                  - a single tensor, if inputs is a single tensor, with
                    exactly the same dimensions as inputs or the first
                    dimension is one and the remaining dimensions match
                    with inputs.
  
                  - a single scalar, if inputs is a single tensor, which will
                    be broadcasted for each input value in input tensor.
  
                  - a tuple of tensors or scalars, the baseline corresponding
                    to each tensor in the inputs' tuple can be:
  
                  - either a tensor with matching dimensions to
                    corresponding tensor in the inputs' tuple
                    or the first dimension is one and the remaining
                    dimensions match with the corresponding
                    input tensor.
  
                  - or a scalar, corresponding to a tensor in the
                    inputs' tuple. This scalar value is broadcasted
                    for corresponding input tensor.
  
                  Default: None
  
          attributions (Tensor or tuple[Tensor, ...]):
                  Attribution scores computed based on an attribution algorithm.
                  This attribution scores can be computed using the implementations
                  provided in the `captum.attr` package. Some of those attribution
                  approaches are so called global methods, which means that
                  they factor in model inputs' multiplier, as described in:
                  https://arxiv.org/abs/1711.06104
                  Many global attribution algorithms can be used in local modes,
                  meaning that the inputs multiplier isn't factored in the
                  attribution scores.
                  This can be done duing the definition of the attribution algorithm
                  by passing `multipy_by_inputs=False` flag.
                  For example in case of Integrated Gradients (IG) we can obtain
                  local attribution scores if we define the constructor of IG as:
                  ig = IntegratedGradients(multipy_by_inputs=False)
  
                  Some attribution algorithms are inherently local.
                  Examples of inherently local attribution methods include:
                  Saliency, Guided GradCam, Guided Backprop and Deconvolution.
  
                  For local attributions we can use real-valued perturbations
                  whereas for global attributions that perturbation is binary.
                  https://arxiv.org/abs/1901.09392
  
                  If we want to compute the infidelity of global attributions we
                  can use a binary perturbation matrix that will allow us to select
                  a subset of features from `inputs` or `inputs - baselines` space.
                  This will allow us to approximate sensitivity-n for a global
                  attribution algorithm.
  
                  `infidelity_perturb_func_decorator` function decorator is a helper
                  function that computes perturbations under the hood if perturbed
                  inputs are provided.
  
                  For more details about how to use `infidelity_perturb_func_decorator`,
                  please, read the documentation about `perturb_func`
  
                  Attributions have the same shape and dimensionality as the inputs.
                  If inputs is a single tensor then the attributions is a single
                  tensor as well. If inputs is provided as a tuple of tensors
                  then attributions will be tuples of tensors as well.
  
          additional_forward_args (Any, optional): If the forward function
                  requires additional arguments other than the inputs for
                  which attributions should not be computed, this argument
                  can be provided. It must be either a single additional
                  argument of a Tensor or arbitrary (non-tuple) type or a tuple
                  containing multiple additional arguments including tensors
                  or any arbitrary python types. These arguments are provided to
                  forward_func in order, following the arguments in inputs.
                  Note that the perturbations are not computed with respect
                  to these arguments. This means that these arguments aren't
                  being passed to `perturb_func` as an input argument.
  
                  Default: None
          target (int, tuple, Tensor, or list, optional): Indices for selecting
                  predictions from output(for classification cases,
                  this is usually the target class).
                  If the network returns a scalar value per example, no target
                  index is necessary.
                  For general 2D outputs, targets can be either:
  
                  - A single integer or a tensor containing a single
                    integer, which is applied to all input examples
  
                  - A list of integers or a 1D tensor, with length matching
                    the number of examples in inputs (dim 0). Each integer
                    is applied as the target for the corresponding example.
  
                    For outputs with > 2 dimensions, targets can be either:
  
                  - A single tuple, which contains #output_dims - 1
                    elements. This target index is applied to all examples.
  
                  - A list of tuples with length equal to the number of
                    examples in inputs (dim 0), and each tuple containing
                    #output_dims - 1 elements. Each tuple is applied as the
                    target for the corresponding example.
  
                  Default: None
          n_perturb_samples (int, optional): The number of times input tensors
                  are perturbed. Each input example in the inputs tensor is expanded
                  `n_perturb_samples`
                  times before calling `perturb_func` function.
  
                  Default: 10
          max_examples_per_batch (int, optional): The number of maximum input
                  examples that are processed together. In case the number of
                  examples (`input batch size * n_perturb_samples`) exceeds
                  `max_examples_per_batch`, they will be sliced
                  into batches of `max_examples_per_batch` examples and processed
                  in a sequential order. If `max_examples_per_batch` is None, all
                  examples are processed together. `max_examples_per_batch` should
                  at least be equal `input batch size` and at most
                  `input batch size * n_perturb_samples`.
  
                  Default: None
          normalize (bool, optional): Normalize the dot product of the input
                  perturbation and the attribution so the infidelity value is invariant
                  to constant scaling of the attribution values. The normalization factor
                  beta is defined as the ratio of two mean values:
  
                  .. math::
                      \beta = \frac{
                          \mathbb{E}_{I \sim \mu_I} [ I^T \Phi(f, x) (f(x) - f(x - I)) ]
                      }{
                          \mathbb{E}_{I \sim \mu_I} [ (I^T \Phi(f, x))^2 ]
                      }
  
                  Please refer the original paper for the meaning of the symbols. Same
                  normalization can be found in the paper's official implementation
                  https://github.com/chihkuanyeh/saliency_evaluation
  
                  Default: False
      Returns:
  
          infidelities (Tensor): A tensor of scalar infidelity scores per
                  input example. The first dimension is equal to the
                  number of examples in the input batch and the second
                  dimension is one.
  
      Examples::
          >>> # ImageClassifier takes a single input tensor of images Nx3x32x32,
          >>> # and returns an Nx10 tensor of class probabilities.
          >>> net = ImageClassifier()
          >>> saliency = Saliency(net)
          >>> input = torch.randn(2, 3, 32, 32, requires_grad=True)
          >>> # Computes saliency maps for class 3.
          >>> attribution = saliency.attribute(input, target=3)
          >>> # define a perturbation function for the input
          >>> def perturb_fn(inputs):
          >>>    noise = torch.tensor(np.random.normal(0, 0.003, inputs.shape)).float()
          >>>    return noise, inputs - noise
          >>> # Computes infidelity score for saliency maps
          >>> infid = infidelity(net, perturb_fn, input, attribution)
      """
  
      def _generate_perturbations(
          current_n_perturb_samples: int,
      ) -> Tuple[
          TensorOrTupleOfTensorsGeneric, TensorOrTupleOfTensorsGeneric
      ]:
          r"""
          The perturbations are generated for each example
          `current_n_perturb_samples` times.
  
          For performance reasons we are not calling `perturb_func` on each example but
          on a batch that contains `current_n_perturb_samples`
          repeated instances per example.
          """
  
          def call_perturb_func():
              r""" """
              baselines_pert = None
              inputs_pert: Union[Tensor, Tuple[Tensor, ...]]
              if len(inputs_expanded) == 1:
                  inputs_pert = inputs_expanded[0]
                  if baselines_expanded is not None:
                      baselines_pert = cast(Tuple, baselines_expanded)[
                          0
                      ]
              else:
                  inputs_pert = inputs_expanded
                  baselines_pert = baselines_expanded
              return (
                  perturb_func(inputs_pert, baselines_pert)
                  if baselines_pert is not None
                  else perturb_func(inputs_pert)
              )
  
          inputs_expanded = tuple(
              torch.repeat_interleave(
                  input, current_n_perturb_samples, dim=0
              )
              for input in inputs
          )
  
          baselines_expanded = baselines
          if baselines is not None:
              baselines_expanded = tuple(
                  baseline.repeat_interleave(
                      current_n_perturb_samples, dim=0
                  )
                  if isinstance(baseline, torch.Tensor)
                  and baseline.shape[0] == input.shape[0]
                  and baseline.shape[0] > 1
                  else baseline
                  for input, baseline in zip(
                      inputs, cast(Tuple, baselines)
                  )
              )
  
          return call_perturb_func()
  
      def _validate_inputs_and_perturbations(
          inputs: Tuple[Tensor, ...],
          inputs_perturbed: Tuple[Tensor, ...],
          perturbations: Tuple[Tensor, ...],
      ) -> None:
          # asserts the sizes of the perturbations and inputs
          assert len(perturbations) == len(inputs), (
              """The number of perturbed
              inputs and corresponding perturbations must have the same number of
              elements. Found number of inputs is: {} and perturbations:
              {}"""
          ).format(len(perturbations), len(inputs))
  
          # asserts the shapes of the perturbations and perturbed inputs
          for perturb, input_perturbed in zip(
              perturbations, inputs_perturbed
          ):
              assert perturb[0].shape == input_perturbed[0].shape, (
                  """Perturbed input
                  and corresponding perturbation must have the same shape and
                  dimensionality. Found perturbation shape is: {} and the input shape
                  is: {}"""
              ).format(perturb[0].shape, input_perturbed[0].shape)
  
      def _next_infidelity_tensors(
          current_n_perturb_samples: int,
      ) -> Union[Tuple[Tensor], Tuple[Tensor, Tensor, Tensor]]:
          perturbations, inputs_perturbed = _generate_perturbations(
              current_n_perturb_samples
          )
  
          perturbations = _format_tensor_into_tuples(perturbations)
          inputs_perturbed = _format_tensor_into_tuples(
              inputs_perturbed
          )
  
          _validate_inputs_and_perturbations(
              cast(Tuple[Tensor, ...], inputs),
              cast(Tuple[Tensor, ...], inputs_perturbed),
              cast(Tuple[Tensor, ...], perturbations),
          )
  
          targets_expanded = _expand_target(
              target,
              current_n_perturb_samples,
              expansion_type=ExpansionTypes.repeat_interleave,
          )
          additional_forward_args_expanded = (
              _expand_additional_forward_args(
                  additional_forward_args,
                  current_n_perturb_samples,
                  expansion_type=ExpansionTypes.repeat_interleave,
              )
          )
  
          inputs_perturbed_fwd = _run_forward(
              forward_func,
              inputs_perturbed,
              targets_expanded,
              additional_forward_args_expanded,
          )
          inputs_fwd = _run_forward(
              forward_func, inputs, target, additional_forward_args
          )
          inputs_fwd = torch.repeat_interleave(
              inputs_fwd, current_n_perturb_samples, dim=0
          )
          perturbed_fwd_diffs = inputs_fwd - inputs_perturbed_fwd
          attributions_expanded = tuple(
              torch.repeat_interleave(
                  attribution, current_n_perturb_samples, dim=0
              )
              for attribution in attributions
          )
  
          attributions_times_perturb = tuple(
              (attribution_expanded * perturbation).view(
                  attribution_expanded.size(0), -1
              )
              for attribution_expanded, perturbation in zip(
                  attributions_expanded, perturbations
              )
          )
  
          attr_times_perturb_sums = sum(
              torch.sum(attribution_times_perturb, dim=1)
              for attribution_times_perturb in attributions_times_perturb
          )
          attr_times_perturb_sums = cast(
              Tensor, attr_times_perturb_sums
          )
  
          # reshape as Tensor(bsz, current_n_perturb_samples)
          attr_times_perturb_sums = attr_times_perturb_sums.view(
              bsz, -1
          )
          perturbed_fwd_diffs = perturbed_fwd_diffs.view(bsz, -1)
  
          if normalize:
              # in order to normalize, we have to aggregate the following tensors
              # to calculate MSE in its polynomial expansion:
              # (a-b)^2 = a^2 - 2ab + b^2
              return (
                  attr_times_perturb_sums.pow(2).sum(-1),
                  (attr_times_perturb_sums * perturbed_fwd_diffs).sum(
                      -1
                  ),
                  perturbed_fwd_diffs.pow(2).sum(-1),
              )
          else:
              # returns (a-b)^2 if no need to normalize
              return (
                  (attr_times_perturb_sums - perturbed_fwd_diffs)
                  .pow(2)
                  .sum(-1),
              )
  
      def _sum_infidelity_tensors(agg_tensors, tensors):
          return tuple(
              agg_t + t for agg_t, t in zip(agg_tensors, tensors)
          )
  
      # perform argument formattings
      inputs = _format_tensor_into_tuples(inputs)  # type: ignore
      if baselines is not None:
          baselines = _format_baseline(
              baselines, cast(Tuple[Tensor, ...], inputs)
          )
      additional_forward_args = _format_additional_forward_args(
          additional_forward_args
      )
      attributions = _format_tensor_into_tuples(attributions)  # type: ignore
  
      # Make sure that inputs and corresponding attributions have matching sizes.
      assert len(inputs) == len(attributions), (
          """The number of tensors in the inputs and
          attributions must match. Found number of tensors in the inputs is: {} and in the
          attributions: {}"""
      ).format(len(inputs), len(attributions))
      for inp, attr in zip(inputs, attributions):
          assert inp.shape == attr.shape, (
              """Inputs and attributions must have
          matching shapes. One of the input tensor's shape is {} and the
          attribution tensor's shape is: {}"""
          ).format(inp.shape, attr.shape)
  
      bsz = inputs[0].size(0)
      with torch.no_grad():
          # if not normalize, directly return aggrgated MSE ((a-b)^2,)
          # else return aggregated MSE's polynomial expansion tensors (a^2, ab, b^2)
          agg_tensors = _divide_and_aggregate_metrics(
              cast(Tuple[Tensor, ...], inputs),
              n_perturb_samples,
              _next_infidelity_tensors,
              agg_func=_sum_infidelity_tensors,
              max_examples_per_batch=max_examples_per_batch,
          )
  
      if normalize:
          beta_num = agg_tensors[1]
          beta_denorm = agg_tensors[0]
  
          beta = safe_div(beta_num, beta_denorm)
  
          infidelity_values = (
              beta**2 * agg_tensors[0]
              - 2 * beta * agg_tensors[1]
              + agg_tensors[2]
          )
      else:
          infidelity_values = agg_tensors[0]
  
      infidelity_values /= n_perturb_samples
  
      return infidelity_values
  
file_path: captum/captum/metrics/_core/infidelity.py
imports: from typing import Any, Callable, cast, Tuple, Union
  import torch
  from captum._utils.common import _expand_additional_forward_args, _expand_target, _format_additional_forward_args, _format_baseline, _format_tensor_into_tuples, _run_forward, ExpansionTypes, safe_div
  from captum._utils.typing import BaselineType, TargetType, TensorOrTupleOfTensorsGeneric
  from captum.log import log_usage
  from captum.metrics._utils.batching import _divide_and_aggregate_metrics
  from torch import Tensor
label_desc: Use image node_id 2 for calling a global function with example usage: infidelity(forward_func, perturb_func, inputs, attributions, baselines, additional_forward_args, target, n_perturb_samples, max_examples_per_batch, normalize) and returns: infidelity_values, call_perturb_func, tuple, unknown, attr_times_perturb_sums,  unknown,  perturbed_fwd_diffs, unknown