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I have a feeling that it's related to how different output layers are stacked up against one another? However, there's a missing link in that argument that I can't completely get to.

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In a convolution neural network, every neuron in layer $i+1$ depends on few neurons in layer $i$. It thus aggregated local information for layer $i$. As you go through the layers, the subset of input that the neuron sees grows, and so the features depend on a larger part of the input. Eventually, neurons depend, in directly, on all inputs.

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