class_docstr: This is a computationally-efficient implementation that computes the type of
  "infinitesimal" influence scored defined in the paper "Understanding Black-box
  Predictions via Influence Functions" by Koh et al
  (https://arxiv.org/pdf/1703.04730.pdf). This implementation does *not* follow
  the approach in that paper, however. Instead, it follows an implementation that is
  several orders of magnitudes faster, described in the paper "Scaling Up Influence
  Functions" by Schioppa et al (https://arxiv.org/pdf/2112.03052.pdf).
  
  This implementation computes a low-rank approximation of the inverse Hessian, i.e.
  a tall and skinny (with width k) matrix :math`R` such that
  :math`H^{-1} \approx RR'`, where k is small. In particular, let :math`V` be the
  matrix of width k whose columns contain the top-k eigenvectors of :math`H`, and let
  :math`S` be the k by k matrix whose diagonals contain the corresponding eigenvalues.
  This implementation lets :math`R=VS^{-0.5}`. Thus, the core computational step is
  computing the top-k eigenvalues / eigenvectors.
  
  This approximation is useful for several reasons:
  - It avoids numerical issues associated with inverting small eigenvalues
  - Since the influence score is given by
    :math`\nabla_\theta L(x)' H^{-1} \nabla_\theta L(z)`, which is approximated by
    :math`(\nabla_\theta L(x)' R) (\nabla_\theta L(z)' R)`, we can compute an
    "influence embedding" for a given example :math`x`, :math`\nabla_\theta L(x)' R`,
    such that the influence score of one example on another is approximately the
    dot-product of their respective embeddings.
  - Even for large models, we can store `R` in memory, provided k is small. This
    means influence embeddings (and thus influence scores) can be efficiently
    computed by doing a backwards pass to compute :math`\nabla_\theta L(x)` and then
    multiplying by :math`R'`. This is orders of magnitude faster than the previous
    LISSA approach of Koh et al, which to compute the influence score involving a
    given example, need to compute Hessian-vector products involving on the order
    of 10^4 examples.
  
  The key novelty of the approach by Schioppa et al is that it uses the Arnoldi
  iteration to find the top-k eigenvalues / eigenvectors of the Hessian without
  explicitly forming the Hessian. In more detail, the approach first runs the
  Arnoldi iteration, which only requires the ability to compute Hessian-vector
  products, to find a Krylov subspace of moderate dimension, i.e. 200. It then finds
  the top-k eigevalues / eigenvectors of the restriction of the Hessian to the
  subspace, where k is small, i.e. 50. Finally, it expresses the eigenvectors in
  the original basis. This approach for finding the top-k eigenvalues / eigenvectors
  is justified by the property of the Arnoldi iteration, that the Krylov subspace
  it returns tends to contain the top eigenvectors.
  
  This implementation require some computation time `__init__`, where it
  runs the Arnoldi iteration to calculate `R`. This computation is linear in
  `arnoldi_dim` as well as the size of `hessian_dataset`. After that initial
  overhead, calculation of influence scores is quick, only requiring a backwards pass
  and multiplication, per example.
  
  Unlike `NaiveInfluenceFunction`, this implementation does not flatten any
  parameters, as the 2D Hessian is never formed, and Pytorch's Hessian-vector
  implementation (`torch.autograd.functional.hvp`) allows the input and output
  vector to be a tuple of tensors. Avoiding flattening / unflattening parameters
  brings scalability gains.
docstr: Args:
  
      inputs (tuple): `inputs` is the test batch and is a tuple of
              any, where the last element is assumed to be the labels for the
              batch. That is, `model(*batch[0:-1])` produces the output for
              `model`, and `batch[-1]` are the labels, if any. This is the same
              assumption made for each batch yielded by training dataset
              `train_dataset` - please see its documentation in `__init__` for
              more details on the assumed structure of a batch.
      show_progress (bool, optional): To compute the influence of examples in
              training dataset `train_dataset`, we compute the influence
              of each batch. If `show_progress` is true, the progress of this
              computation will be displayed. In particular, the number of batches
              for which influence has been computed will be displayed. It will
              try to use tqdm if available for advanced features (e.g. time
              estimation). Otherwise, it will fallback to a simple output of
              progress.
              Default: False
  
  Returns:
      influence_scores (Tensor): Influence scores over the entire
              training dataset `train_dataset`. Dimensionality is
              (inputs_batch_size, src_dataset_size). For example:
              influence_scores[i][j] = the influence score for the j-th training
              example to the i-th example in the test dataset.
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code: def _influence(
      self,
      inputs: Union[Tuple[Any, ...], DataLoader],
      show_progress: bool = False,
  ) -> Tensor:
      r"""
      Args:
  
          inputs (tuple): `inputs` is the test batch and is a tuple of
                  any, where the last element is assumed to be the labels for the
                  batch. That is, `model(*batch[0:-1])` produces the output for
                  `model`, and `batch[-1]` are the labels, if any. This is the same
                  assumption made for each batch yielded by training dataset
                  `train_dataset` - please see its documentation in `__init__` for
                  more details on the assumed structure of a batch.
          show_progress (bool, optional): To compute the influence of examples in
                  training dataset `train_dataset`, we compute the influence
                  of each batch. If `show_progress` is true, the progress of this
                  computation will be displayed. In particular, the number of batches
                  for which influence has been computed will be displayed. It will
                  try to use tqdm if available for advanced features (e.g. time
                  estimation). Otherwise, it will fallback to a simple output of
                  progress.
                  Default: False
  
      Returns:
          influence_scores (Tensor): Influence scores over the entire
                  training dataset `train_dataset`. Dimensionality is
                  (inputs_batch_size, src_dataset_size). For example:
                  influence_scores[i][j] = the influence score for the j-th training
                  example to the i-th example in the test dataset.
      """
      # turn inputs and targets into a dataset. inputs has already been processed
      # so that it should always be unpacked
      inputs_dataset = _format_inputs_dataset(inputs)
      return (
          _influence_helper_intermediate_quantities_influence_function(
              self, inputs_dataset, show_progress
          )
      )
  
imports: import functools
  from typing import Any, Callable, List, Optional, Tuple, Union
  import torch
  from captum._utils.gradient import _extract_parameters_from_layers
  from captum.influence._utils.common import _compute_batch_loss_influence_function_base, _compute_jacobian_sample_wise_grads_per_batch, _dataset_fn, _format_inputs_dataset, _functional_call, _get_k_most_influential_helper, _influence_batch_intermediate_quantities_influence_function, _influence_helper_intermediate_quantities_influence_function, _influence_route_to_helpers, _load_flexible_state_dict, _parameter_add, _parameter_dot, _parameter_linear_combination, _parameter_multiply, _parameter_to, _params_to_names, _progress_bar_constructor, _self_influence_helper_intermediate_quantities_influence_function, _top_eigen, KMostInfluentialResults
  from captum.log import log_usage
  from torch import Tensor
  from torch.nn import Module
  from torch.utils.data import DataLoader, Dataset
  from tqdm import tqdm
  from .influence_function import _get_dataset_embeddings_intermediate_quantities_influence_function, InfluenceFunctionBase, IntermediateQuantitiesInfluenceFunction
label_desc: Use image node_id 6 for calling the ArnoldiInfluenceFunction obj's underlying member method code with example usage: obj._influence(inputs, show_progress) and returns: _influence_helper_intermediate_quantities_influence_function