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When calculating gradients for backpropagation, a surprising result is that the gradient of the weights in a ReLU-activated layer ends up being a diagonal matrix. I am seeking to understand how this makes sense intuitively, in terms of gradient descent.

Does this mean that weights in the non-diagonal positions don't have an impact on the loss? How could that be? Do these weights never get updated at all?

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    $\begingroup$ Think of what what's a gradient of the linear activation $\endgroup$ – Aksakal Feb 28 '18 at 2:37
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a surprising result is that the gradient of the weights in a ReLU-activated layer ends up being a diagonal matrix

Well, it doesn't. You just initialized your weights with a diagonal matrix.

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