I was wondering if there is any value to creating combined features out of multiple categorical variables when the individual categorical variables are already one-hot encoded?
Simple example: there is a variable P with categories {X,Y} and and a variable Q with categories {Z,W}. After one-hot we would have 4 variables, P.X, P.Y, Q.Z and Q.W.
In this scenario I'm wondering if the algorithm (Xgboost or a deep neural network) would sufficiently learn interaction effects between these or is there further value to creating variables: X.Z, X.W, Y.Z, Y.W which would be the unique combinations of P and Q.
The reason I am asking is to try to asses whether to embark on creating these interaction variables in my real-world scenario, where I have 7 such categorical features and 6-15 categories each which would mean thousands of new variables to account for all possible levels of permutations.