code: def frechet_distance(mu_x, sigma_x, mu_y, sigma_y):
      r"""Computes the Fréchet distance between two multivariate normal distributions :cite:`dowson1982frechet`.
  
      Concretely, for multivariate Gaussians :math:`X(\mu_X, \Sigma_X)`
      and :math:`Y(\mu_Y, \Sigma_Y)`, the function computes and returns :math:`F` as
  
      .. math::
          F(X, Y) = || \mu_X - \mu_Y ||_2^2
          + \text{Tr}\left( \Sigma_X + \Sigma_Y - 2 \sqrt{\Sigma_X \Sigma_Y} \right)
  
      Args:
          mu_x (torch.Tensor): mean :math:`\mu_X` of multivariate Gaussian :math:`X`, with shape `(N,)`.
          sigma_x (torch.Tensor): covariance matrix :math:`\Sigma_X` of :math:`X`, with shape `(N, N)`.
          mu_y (torch.Tensor): mean :math:`\mu_Y` of multivariate Gaussian :math:`Y`, with shape `(N,)`.
          sigma_y (torch.Tensor): covariance matrix :math:`\Sigma_Y` of :math:`Y`, with shape `(N, N)`.
  
      Returns:
          torch.Tensor: the Fréchet distance between :math:`X` and :math:`Y`.
      """
      if len(mu_x.size()) != 1:
          raise ValueError(
              f"Input mu_x must be one-dimensional; got dimension {len(mu_x.size())}."
          )
      if len(sigma_x.size()) != 2:
          raise ValueError(
              f"Input sigma_x must be two-dimensional; got dimension {len(sigma_x.size())}."
          )
      if sigma_x.size(0) != sigma_x.size(1) != mu_x.size(0):
          raise ValueError(
              "Each of sigma_x's dimensions must match mu_x's size."
          )
      if mu_x.size() != mu_y.size():
          raise ValueError(
              f"Inputs mu_x and mu_y must have the same shape; got {mu_x.size()} and {mu_y.size()}."
          )
      if sigma_x.size() != sigma_y.size():
          raise ValueError(
              f"Inputs sigma_x and sigma_y must have the same shape; got {sigma_x.size()} and {sigma_y.size()}."
          )
  
      a = (mu_x - mu_y).square().sum()
      b = sigma_x.trace() + sigma_y.trace()
      c = torch.linalg.eigvals(sigma_x @ sigma_y).sqrt().real.sum()
      return a + b - 2 * c
  
imports: import math
  import tempfile
  import warnings
  from collections.abc import Sequence
  from typing import List, Optional, Tuple, Union
  import torch
  import torchaudio
  from torch import Tensor
  from torchaudio._internal.module_utils import deprecated
  from .filtering import highpass_biquad, treble_biquad
label_desc: Use image node_id 56 for calling a global function with example usage: frechet_distance(mu_x, sigma_x, mu_y, sigma_y) and returns: unknown