In matrix completion problem,
$$\min_{X} \mbox{rank}(X)~~s.t.~~x_{ij}=y_{ij},~\forall ij \in \Omega$$
where $X$ denotes the matrix to recover, $\Omega$ is the set of the known entries and $y_{ij}$ are their values. This rank-minimization problem ought to be NP-hard. So, replace the rank with the nuclear norm and use the Hadamard product,
$$\min_X \|X\|_* + \frac{\mu}{2} \|\Omega \circ(X-Y)\|_F^2$$
But why is this expression reduced to being transformed as above? And please tell me more about this formula. Thank you.