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We define the rank of a matrix as the minimal number of rank-1 (i.e. of the form w^\dagger{}v) matrices that sum to it. I wonder what happens when we decide to optimize for something else: say, norm of type $foo over norms of type $bar over those matrices. Obvious candidates for $foo are L_{something}, for $bar are operator norms or Frobenius norm.
In particular, $foo=L_n and $bar=operator norm inherited from L_n seems potentially interesting (in particular for n=2).
