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When training a neural network, I appreciate that data normalisation helps training. However, is it a good idea to normalise the data in multiple ways. For instance, is it a good idea to apply z-score normalisation on min-max normalised data? That is if the input data is already normalised to [0, 1], is it a good idea to train on the z-scores of that?

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I've not seen any paper about that but based on what I've faced till now, normalizing data intuitively is just for assigning same importance to different features which their raw values do not have a same range. Take a look at here. Also, you can take a look at here that professor says that you just need to employ a technique and it's not really important which technique. Also, take a look at here.

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Depends on the dataset, and what you're looking for.

It is possible to do multiple data transformations. Standardization is often the safest choice, but only when the data is normally distributed.

If you have normalized the data, you can always apply standardisation. Standardization does not mutate the data set, but rescaling does create a new data set. From the standardized data set you can go back to the original data set.

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