Data augmentation is one of the ugliest hacks in ML. If you know what the invariances are, encode them into the architecture. Don't blow up the size of you dataset in order to approximate them.
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Perhaps the emphasis should be on learning the invariances from the data rather than encoding assumed invariances through architecture or augmentation?
Sure, for invariances you don't know a priori. For the ones you do, learning them is just a waste of time.
