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I am trying to implement the back-propagation of a simple convolutional network. Specifically I understand that one of the steps is the convolution of the gradients coming from the next layer, with the rotated kernels. However I cannot compute the parameters of this convolution. Let's see this with an example:

Imagine that the layer's input is an image of size (heigh, width) == (4, 4) with only 1 color channel. We also use zero-padding == 2 and stride == 1. Our kernel (let's say we only have 1 kernel) is of size (filterHeight, filterWidth) == (2, 2). We can now use the following equation to compute each dimension of the output:

out = (in - filter + 2*padding) / stride + 1 (1)

So the output volume has a (height, width) == (7, 7) and everything is fine.

Now comes the time for the backward pass. I need to convolve this (7, 7) tensor with the rotated (2, 2) weights and get a tensor of (4, 4). How can I compute the stride and padding? I only have one equation (1) but two unknowns! Therefore there isn't a single solution for stride and padding!

I can heuristically find that the combination of padding == 2 and stride == 3 will work out, but why choose this over the potentially infinite number of valid combinations?

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As far as I know you would need to perform a "full" convolution during the backpropagation step. So the gradients from the l+1 layer will be a (7, 7) tensor. The "full" convolution with the rotated filter (2, 2) will result in a (8, 8) tensor. Removing the original padding as added to your input (4, 4) would again result in a (4, 4) tensor. There would be no need to calculate or guess any further padding/stride while doing the backprop convolution.

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