Very often, data augmentation is applied only to classification models, so that the inputs change but the target outputs (“labels”) don’t change.

I’m primarily interested in regression models; in these cases, typically when one augments the input, one also has to modify the target outputs appropriately.

I’d love to hear more in the future about augmenting for regression problems too.

It seems like there’s an important, ‘useful’ class of transformations T which may satisfy the following relation:

Given a (nonlinear) function f mapping inputs x to outputs y, i.e. y = f(x), some transformation T may satisfy T(y) = f(T(x)), i.e. that T and f commute under composition: T(f(x)) = f(T(x)).

(This will of course depend on the form of f(x), which is usually unknown – that’s why you’re using a neural network to approximate it).

Often, translation in time or space will satisfy this property, as will inversion (sometimes), depending on the data & function…

**Question:** Is there a “name” for this type of mathematical transformation/symmetry, and perhaps a *mechanism* to find such ‘allowed’ transformations?

(perhaps something akin to a ‘Calculus of Variations’ approach could work…)