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Is there a term for using the best and worst results of a neural network classifier as labelled input for another neural network? Or isn't this approach valid at all?

E.g. I train a neural network and then I classify unlabeled samples with this neural network. The outputs I use a labelled training for a new neural network.

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If you use the new predicted outputs combined with your known labelled samples, this could be considered self-training, a form of semi-supervised learning. See for example [1], slide 19. As is any form of semi-supervised learning available today, whether it's a valid approach depends on whether its assumptions hold in your problem (in this case whether your trust in the predictions is justified).

[1] http://pages.cs.wisc.edu/~jerryzhu/pub/sslicml07.pdf

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Ofcourse you can do it. But I am not sure it will get you to the answer that you desire.

Instead, I would recommend that you use the input vectors in NN2 as input vectors to NN1.

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