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I am wondering about how 1x1 convolution can be used to change the dimensionality of feature maps in a residual learning network.enter image description here

Here the top 1x1 convolution changes the feature map size from 256 to 64. How is this possible?

In a previous post explaining 1x1 convolution in neural net, it is mentioned that, if a layer having $n_1$ feature maps is subjected to 1x1 convolution with $n_2$ filters then number of feature map changes to $n_2$. Shouldn't it be $n_1$$n_2$ since each of the $n_2$ filters produce one output corresponding to each of the $n_1$ inputs.

Also how does one generate 256 feature maps from 64, as done in the bottom layer.

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There's only one parameter for each input map in a 1*1 filter, actually the 1*1 convolution is multiplying the every element of an input map by the same scalar.

So it is similar to getting 265 linear combinations out of 64 variables, the $n$-th feature map $y_n$ is like,

$$y_n=f(w_{n,1}x_1+w_{n,2}x_2+...+w_{n,64}x_{64})$$ so actually we can get any number of output feature maps as we want. Of course if the output dimension is greater than the input dimension, the output would be redundant.

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  • $\begingroup$ What is $w^n_i$ ?. Is it just $w_i$ $\endgroup$
    – Newstein
    Jul 1, 2016 at 9:50
  • $\begingroup$ @Newstein It's meant to be in accordance with the subscript of $y_n$, so you have a different set of weights for different output channels. $\endgroup$
    – dontloo
    Jul 1, 2016 at 9:53
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    $\begingroup$ @Newstein yes by weights I mean the single parameter in a 1x1 filter, and there're not in total 64 filters, there are 64 output channels (or feature maps), which means there'll be 64*256 filters (you apply a different filter to each input channel, add up the results and apply an activation function to get the output of one channel) $\endgroup$
    – dontloo
    Jul 1, 2016 at 10:37
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    $\begingroup$ I really think this answer is misleading. There a 1x1 convolutional filter is "1x1" just in the spatial dimension, but it is a 1x1xC convolution, where C is the number of input feature maps or channels (256 for the first layer in the OPs example example). Therefore, you actually have C parameters for a 1x1 convolution. Since you want to output 64 different feature maps, you have 1x1xCx64 parameters for the whole layer. $\endgroup$
    – Pepe Mandioca
    Feb 8, 2018 at 16:47
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    $\begingroup$ @facuq yea it is, I've made some edit $\endgroup$
    – dontloo
    Feb 9, 2018 at 3:59

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