How to select 500 most pertinents tags among 10000? Say we have 100,000 documents tagged with 10,000 different tags (Max 5 tag per document). We wish to limit allowed tags to a list of 500 tags. 

How to select 500 tags in order to cover the largest set of documents ?

Firstly, I chose the 500 most frequent tags. If we keep only these 500 tags, 70% of documents keep at least 1 tag.

I'm looking for a better method to pick the 500 most pertinents tags.

example : 

my best set of 5 tags to cover all documents is [italian, german, french, chineese, japan]. 

it is not [italian, spanish, indian, finnish, english] although these are the most frequent tags.

I tried a Single Value Decomposition on the matrix : [documentID, set of tags]. but after that ? is it a good idea ? how to get 500 tags from the SVD results ?
That is a set cover problem. The documents (universe) are being covered by a set of tags (elements). 

In the case of maximizing the number of documents with a limited number of tags, a greedy approach can be taken. Find the single tag that covers the most documents. Then find the next tag that covers the most documents not already covered. Continue the process until the number of tags has been reached.