Understanding Convolution with an Even Sized Filter

If I have an even sized convolutional filter, say 2 by 2, f_{i,j} for 1<= i <=2, 1<= j < = 2

Then if I apply it to a window centered at (x,y) (assuming appropriate padding),
then will it apply something like
f_{1,1} (x,y) + f_{1,2} (x+1, y) + f_{2, 1}(x, y+1) + f_{2,2}(x+1, y+1) in Pytorch?
I’m not sure what the formula would be for even sized filters basically.
Thanks.

I’m not sure what your notation means exactly, but you could use a simple example as given here:

weight = torch.ones(1, 1, 2, 2)
x = torch.arange(4*4).view(1, 1, 4, 4).float()
print(weight)
> tensor([[[[1., 1.],
            [1., 1.]]]])

print(x)
> tensor([[[[ 0.,  1.,  2.,  3.],
            [ 4.,  5.,  6.,  7.],
            [ 8.,  9., 10., 11.],
            [12., 13., 14., 15.]]]])

out = F.conv2d(x, weight, stride=1, padding=0)
print(out)
> tensor([[[[10., 14., 18.],
            [26., 30., 34.],
            [42., 46., 50.]]]])

out = F.conv2d(x, weight, stride=1, padding=1)
print(out)
> tensor([[[[ 0.,  1.,  3.,  5.,  3.],
            [ 4., 10., 14., 18., 10.],
            [12., 26., 30., 34., 18.],
            [20., 42., 46., 50., 26.],
            [12., 25., 27., 29., 15.]]]])

As you can see, the kernel will use windows of 2x2 to create its output.

Let me know, if this example makes the output clear or if you would need more information.