To implement bilinear interpolation upsampling using a transposed convolutional layer, the layer's kernel weights are explicitly initialized using a bilinear kernel function. This ensures the layer performs the equivalent distance-weighted upsampling. For instance, to double the height and width of an image with $$3$$ channels, a transposed convolutional layer is created with a stride of $$2$$, padding of $$1$$, and a kernel size of $$4$$. Its weights are then populated by copying the output of the `bilinear_kernel` function.

```python
# PyTorch
conv_trans = nn.ConvTranspose2d(3, 3, kernel_size=4, padding=1, stride=2,
                                bias=False)
conv_trans.weight.data.copy_(bilinear_kernel(3, 3, 4))
```

```python
# MXNet
conv_trans = nn.Conv2DTranspose(3, kernel_size=4, padding=1, strides=2)
conv_trans.initialize(init.Constant(bilinear_kernel(3, 3, 4)))
```

Claude

The upsampling operation of bilinear interpolation can be implemented using a transposed convolutional layer by initializing its weights with a specially constructed bilinear kernel. A programmatic implementation of this kernel calculates weights based on relative distances from the center. For example, the `bilinear_kernel` function constructs a kernel tensor where the values decrease linearly from the center, effectively mimicking the distance-weighted averaging of bilinear interpolation.

```python
def bilinear_kernel(in_channels, out_channels, kernel_size):
    factor = (kernel_size + 1) // 2
    if kernel_size % 2 == 1:
        center = factor - 1
    else:
        center = factor - 0.5
    og = (torch.arange(kernel_size).reshape(-1, 1),
          torch.arange(kernel_size).reshape(1, -1))
    filt = (1 - torch.abs(og[0] - center) / factor) * \
           (1 - torch.abs(og[1] - center) / factor)
    weight = torch.zeros((in_channels, out_channels,
                          kernel_size, kernel_size))
    weight[range(in_channels), range(out_channels), :, :] = filt
    return weight
```

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