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I am trying to understand the purpose of the max pooling layers that are insterted between intermediate convolutional layers. As we know, the outputs of convolutional layers are translational equivariant. That is, if a filter has learn to detect a nose, if the nose is shifted the activation in the resulting feature map will also be shifted by the same amount.

So, we can scan the output of Conv(n) layer with the filters from the Conv(n+1) layer and still detect the relevant features. Then, we can use a global pooling operation after the final Conv layer to make our architecture translation invariant. What is the purpose of the Max Pooling layer then?

In the Deep Learning Book, it is stated that:

Invariance to local translation can be a very useful property if we care more about whether some feature is present than exactly where it is.

Can't we achieve this by just using a Global Pooling layer after the final Conv layer?

I add the following image from the Hands-on Machine Learning with Scikit-Learn, Keras, and TensorFlow from which I can see a potential disadvantage of not using Max Pooling layers:

enter image description here

Suppose that the black pixels (lets say black pixel means 1 and white pixel means 0) on the output of the Max Pooling correspond to some detected feature. Then, the output of next Conv layer, assuming a filter with values and a stride of 2: $$ \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} $$

Conv output (assuming left Max Pooling) $$\begin{bmatrix} 2 & 2 \\ 2 & 2 \\ \end{bmatrix} $$

Conv output (assuming mid Max Pooling) $$\begin{bmatrix} 2 & 2 \\ 2 & 2 \\ \end{bmatrix} $$

Conv output (assuming right Max Pooling) $$\begin{bmatrix} 0 & 4 \\ 0 & 4 \\ \end{bmatrix} $$

Intuitively, the problem I see is that by not being translation invariant the following layers after this Conv layer will receive different outputs and this will make their job harder during training since they have to readjust the values of their filter weights depending on where exactly the feature was detected. Is this intuition correct?

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Intuitively, the problem I see is that by not being translation invariant the following layers after this Conv layer will receive different outputs and this will make their job harder during training since they have to readjust the values of their filter weights depending on where exactly the feature was detected. Is this intuition correct?

This intuition is correct. We need local translation invariance achieved by applying small pooling layers. Global pooling is an overkill for every pooling layer. It guarantees invariance but gets rid of many other useful interactions that could be learnt by the next convolutional layers.

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  • $\begingroup$ Neverthless is not overkill after the final convolutional layer right? $\endgroup$
    – ado sar
    Jul 15 at 12:26
  • $\begingroup$ Well, as you said, it's the final layer, and we expect the network to summarize the information up until that point. But, if you do that in the first conv. layer, how is that going to be possible? $\endgroup$
    – gunes
    Jul 15 at 13:36
  • $\begingroup$ Of course, that's why I said in the OP that the Global Pooling should be after the final Conv layer. $\endgroup$
    – ado sar
    Jul 15 at 15:14
  • $\begingroup$ Maybe I need to ask clarifications for this: In response to "Invariance to local translation can be a very useful property if we care more about whether some feature is present than exactly where it is.", you ask "Can't we achieve this by just using a Global Pooling layer after the final Conv layer?" So, what I understand from these two sentences is we drop the intermediate max pooling layers, and just use a global pool layer after the final convolution, and we still get the translation invariance. Isn't this what you mean? $\endgroup$
    – gunes
    Jul 15 at 15:24
  • $\begingroup$ Apologize if I was misunderstood. Yes, this is what I mean. $\endgroup$
    – ado sar
    Jul 15 at 15:30

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