Why, in binary classification, would we map only from the input to the feature space?

Question

I am following these slides on NLP and machine learning. On slide 7, the author says "in binary classification we can map only from the input to the feature space." This seems to be different from non-binary classification (presented earlier in the slide) where you map from X x Y to the feature space. Why for binary classification do you not map from all possible combinations of X x Y? It would seem like each possible X could get assigned one of two labels from Y (i.e. X x Y -->r^n)

Answer 1

In short - these slides are missleading. You can treat the binary classification as multi-label classification and so no additional restrictions apply. However, the trick with X x Y -> F is simply redundant in binary classification. As here everything that gives you any information about classifing to class 0 gives you information about classification to class 1 also (as there is no other option, only two possibilities), while in multi class scenario not being a part of class 0 gives you no actual information (it can still be the part of class 2 or k) so there is a reason behind defining features just for some classes. To sum up:

• Despite what is written in these slides you can treat binary classification as multi-class one
• Using X x Y -> F mapping in binary classification is redundant