What are the downsides of modelling a multi-label problem as a multi-class problem with a single classifier?
Let my clarify what I mean.
There at least two ways that one multi-label problem can be transformed to a multi-class problem with a single classifier (suppose there are N labels at our problem):
Create a class for each element of the powerset of the labels.
Therefore, each element for each combination of the labels.
In this case, the output vector will have $2^N$ length.
Have an output vector of $N$ length (every element of the vector will be one label) but the problem will be treated as a multiclass problem with one classifier.
In this case, let's say the classes which have output probability more than 0.2 will be considered as the classes/labels of this instance/observation.
Obviously, the output probabilities of all the classes should sum up to 1.
What are the implications of transforming a multi-label problem to multi-class problem at each of these cases?