How can I get the value at a non-integer position in PyTorch?

In PyTorch, you cannot directly access tensor elements using non-integer indices. Tensors are indexed using integers, representing specific positions in the data structure. However, you can achieve the effect of accessing values at non-integer positions by using interpolation techniques.

Here's a breakdown of how you can accomplish this using interpolation:

1. Linear Interpolation (for 1D Tensors):

For a 1D tensor, you can perform linear interpolation between the two nearest integer indices. Given a non-integer index `x`, you would take the floor `x_floor` (the integer immediately below `x`) and the ceiling `x_ceil` (the integer immediately above `x`). Then you perform a weighted sum of the tensor values at those indices.

Here's an example using PyTorch:

import torch

def linear_interpolation_1d(tensor, x):
  x_floor = torch.floor(x).long()
  x_ceil = torch.ceil(x).long()
  fraction = x - x_floor
  value = (1 - fraction) tensor[x_floor] + fraction tensor[x_ceil]
  return value

tensor = torch.tensor([1.0, 2.0, 3.0, 4.0, 5.0])
non_int_index = 2.5
result = linear_interpolation_1d(tensor, torch.tensor(non_int_index))
print(result) # Output: tensor(3.5000)

2. Bilinear Interpolation (for 2D Tensors):

For a 2D tensor, you would need to use bilinear interpolation. This approach extends linear interpolation to two dimensions. Given a non-integer (x,y) coordinate, you'd interpolate between the four nearest integer positions in the grid.

Here's how you can approach it in PyTorch (note that this is a simplification and using grid_sample is generally recommended for more robust scenarios):

import torch

def bilinear_interpolation_2d(tensor, x, y):
  x_floor = torch.floor(x).long()
  x_ceil = torch.ceil(x).long()
  y_floor = torch.floor(y).long()
  y_ceil = torch.ceil(y).long()
  x_fraction = x - x_floor
  y_fraction = y - y_floor

  value_top_left = tensor[y_floor, x_floor]
  value_top_right = tensor[y_floor, x_ceil]
  value_bottom_left = tensor[y_ceil, x_floor]
  value_bottom_right = tensor[y_ceil, x_ceil]

  value_top = (1 - x_fraction) value_top_left + x_fraction value_top_right
  value_bottom = (1 - x_fraction) value_bottom_left + x_fraction value_bottom_right

  result = (1 - y_fraction) value_top + y_fraction value_bottom
  return result

tensor_2d = torch.tensor([[1.0, 2.0, 3.0],
  [4.0, 5.0, 6.0],
  [7.0, 8.0, 9.0]])
non_int_x = 1.5
non_int_y = 1.5
result_2d = bilinear_interpolation_2d(tensor_2d, torch.tensor(non_int_x), torch.tensor(non_int_y))
print(result_2d) # Output: tensor(5.0000)

3. Using `torch.nn.functional.grid_sample`:

For more complex situations, PyTorch provides `torch.nn.functional.grid_sample`. This function can handle various types of interpolation and transformations. It's designed for use in spatial transformation and resampling scenarios but can be leveraged to get values at non-integer positions by setting up the grid appropriately.

Important Considerations:

- Boundary Handling: The above approaches do not explicitly handle out-of-bounds indices, so be careful with the indices you provide. You will need to ensure that your `x_floor` and `x_ceil`, and `y_floor` and `y_ceil` are all valid indices for your tensor. - Performance: For performance-critical tasks, consider whether your calculations can be vectorized or if using a library optimized for interpolation is beneficial. In many cases, `grid_sample` is a good option because its implementation is generally very efficient. - Higher Dimensions: For higher-dimensional tensors, you can extend these concepts to trilinear interpolation or higher-order interpolation, but this becomes more complex.

In summary, you cannot directly access elements with non-integer indices, but you can simulate that behavior by using interpolation methods such as linear or bilinear interpolation. For complex scenarios, consider using the `grid_sample` function.

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