Transform multi-label problem to multi-class problem 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):

1)

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.

2)

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?
A multi-label problem is when an instance can have several labels, for instance a system which classifies news articles by topic could do this:


instance 1: politics, society
instance 2: sports
instance 3: culture, sports
instance 4: society
...


To turn this into a multiclass problem and still do the exact same task, one needs to create one class for each possible subset existing in the data, for instance:

{ politics-society, sports, culture-sports, society, ...}


If the original multi-labels problem contains $N$ labels, the number of classes in the multiclass problem is $2^N$ in the worst case (number of partitions of the set). 

The main problem is that the classifier needs a representative sample of every class in order to perform well. The classes sports and culture-sports (for example) are now independent from the perspective of the classifier, so the class sport cannot benefit anymore from the instances belonging to culture-sports, as it would be the case in the multi-label problem. 

So in general one would need many more instances to train a multiclass classifier than a multi-label classifier for doing the same thing.