code: def histogram2d(
      x1: torch.Tensor,
      x2: torch.Tensor,
      bins: torch.Tensor,
      bandwidth: torch.Tensor,
      epsilon: float = 1e-10,
  ) -> torch.Tensor:
      """Estimate the 2d histogram of the input tensor.
  
      The calculation uses kernel density estimation which requires a bandwidth (smoothing) parameter.
  
      Args:
          x1: Input tensor to compute the histogram with shape :math:`(B, D1)`.
          x2: Input tensor to compute the histogram with shape :math:`(B, D2)`.
          bins: The number of bins to use the histogram :math:`(N_{bins})`.
          bandwidth: Gaussian smoothing factor with shape shape [1].
          epsilon: A scalar, for numerical stability. Default: 1e-10.
  
      Returns:
          Computed histogram of shape :math:`(B, N_{bins}), N_{bins})`.
  
      Examples:
          >>> x1 = torch.rand(2, 32)
          >>> x2 = torch.rand(2, 32)
          >>> bins = torch.torch.linspace(0, 255, 128)
          >>> hist = histogram2d(x1, x2, bins, bandwidth=torch.tensor(0.9))
          >>> hist.shape
          torch.Size([2, 128, 128])
      """
  
      _, kernel_values1 = marginal_pdf(
          x1.unsqueeze(2), bins, bandwidth, epsilon
      )
      _, kernel_values2 = marginal_pdf(
          x2.unsqueeze(2), bins, bandwidth, epsilon
      )
  
      pdf = joint_pdf(kernel_values1, kernel_values2)
  
      return pdf
  
imports: from typing import Optional, Tuple
  import torch
label_desc: Use image node_id 4 for calling a global function with example usage: histogram2d(x1, x2, bins, bandwidth, epsilon) and returns: pdf