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It is a well-known math fact that composition of linear/affine transformations is still linear/affine. For a naive example,

$\textbf{A}_1\textbf{A}_2\textbf{x}$ is simply $\textbf{A}\textbf{x}$ where $\textbf{A}=\textbf{A}_1\textbf{A}_2\textbf{x}$

Any one knows why in practice multiple linear layers tend to work better, even though it is mathematically equivalent to a single linear layer? Any reference is appreciated!

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    $\begingroup$ Are you sure those "multiple linear layers" don't have nonlinear activations between them? $\endgroup$ – user76284 Oct 10 '19 at 1:54
  • $\begingroup$ can you post a link to a source that explains the result you mention. There is some confusion as to whether or not you mean that a linear network performs better than its "mathematically reduced" network or if you misunderstand network activations. So a link to a source would be helpful. $\endgroup$ – respectful Oct 10 '19 at 3:41
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The key is that the layers of neurons in neural networks are not affine transformations. All commonly used neurons have some kind of non-linearity. The simplest of these is the Rectified Linear Unit (ReLU), which takes the form $y = x$ when $x > 0$ and $y = 0$ for all other values, where $x$ is a weighted sum of the inputs to the neuron.

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  • $\begingroup$ I'm wondering if he read somewhere that linear networks perform better than the reduced architecture. Perhaps someone did an experiment with a linear network and its reduced equivalent and discovered that in practice the network performs better than the reduced form. I asked OP to clarify. $\endgroup$ – respectful Oct 10 '19 at 3:44
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    $\begingroup$ @respectful Maybe. I think a lot of modern presentations show a linear algebraic presentation of neutral networks, and that for a new reader, it can be easy to miss the activation functions and their roles. $\endgroup$ – John Doucette Oct 10 '19 at 13:01

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