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Is there any need to use a non-linear activation function (ReLU, LeakyReLU, Sigmoid, etc.) if the result of the convolution layer is passed through the sliding window max function, like max-pooling, which is non-linear itself? What about the average pooling?

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Let's first recapitulate why the function that calculates the maximum between two or more numbers, $z=\operatorname{max}(x_1, x_2)$, is not a linear function.

A linear function is defined as $y=f(x) = ax + b$, so $y$ linearly increases with $x$. Visually, $f$ corresponds to a straight line (or hyperplane, in the case of 2 or more input variables).

If $z$ does not correspond to such a straight line (or hyperplane), then it cannot be a linear function (by definition).

Let $x_1 = 1$ and let $x_2 \in [0, 2]$. Then $z=\operatorname{max}(x_1, x_2) = x_1$ for all $x_2 \in [0, 1]$. In other words, for the sub-range $x_2 \in [0, 1]$, the maximum between $x_1$ and $x_2$ is a constant function (a horizontal line at $x_1=1$). However, for the sub-range $x_2 \in [1, 2]$, $z$ correspond to $x_2$, that is, $z$ linearly increases with $x_2$. Given that max is not a linear function in a special case, it can't also be a linear function in general.

Here's a plot (computed with Wolfram Alpha) of the maximum between two numbers (so it is clearly a function of two variables, hence the plot is 3D).

enter image description here

Note that, in this plot, both variables, $x$ and $y$, can linearly increase, as opposed to having one of the variables fixed (which I used only to give you a simple and hopefully intuitive example that the maximum is not a linear function).

In the case of convolution networks, although max-pooling is a non-linear operation, it is primarily used to reduce the dimensionality of the input, so that to reduce overfitting and computation. In any case, max-pooling doesn't non-linearly transform the input element-wise.

The average function is a linear function because it linearly increases with the inputs. Here's a plot of the average between two numbers, which is clearly a hyperplane.

enter image description here

In the case of convolution networks, the average pooling is also used to reduce the dimensionality.

To answer your question more directly, the non-linearity is usually applied element-wise, but neither max-pooling nor average pooling can do that (even if you downsample with a $1 \times 1$ window, i.e. you do not downsample at all).

Nevertheless, you don't necessarily need a non-linear activation function after the convolution operation (if you use max-pooling), but the performance will be worse than if you use a non-linear activation, as reported in the paper Systematic evaluation of CNN advances on the ImageNet (figure 2).

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  • $\begingroup$ Is the same not true for ReLU? And even more so for LeakyReLU? $\endgroup$
    – Kasia
    Feb 10, 2020 at 0:10
  • $\begingroup$ @Kasia What exactly are you referring to? $\endgroup$
    – nbro
    Feb 10, 2020 at 0:12
  • $\begingroup$ ReLU is simply a max function. Couldn't it basically be considered a max-pool with additional constant dimension? $\endgroup$
    – Kasia
    Feb 10, 2020 at 0:31
  • $\begingroup$ @Kasia In max-pooling, conceptually, you slide a window. When you apply ReLU you do not slide any window. It's true that ReLU is a max between 0 and the input, but I fail to understand how you want to make it a pooling operation. $\endgroup$
    – nbro
    Feb 10, 2020 at 1:34
  • $\begingroup$ I think the OP is confused about the apparent piecewise linearity of pooling and ReLu. Although relu is partly linear pooling is not since the node selected maybe exchanged. $\endgroup$
    – user9947
    Feb 10, 2020 at 5:08

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