The nuclear norm minimization for the matrix completion problem is given by

\begin{align} \textrm{minimize } \quad &\|X\|_{*}\\ \textrm{subject to } \quad & X_{ij}=M_{ij} \quad \forall (i,j)\in \Omega \end{align}

where $\Omega$ is the set of randomly sampled entries. There are results that show that one can recover the underlying matrix $M$ with high probability given "enough" measurements (e.g., see Theorem 1.1 of Candes and Recht, 2008 paper).

Let the underlying matrix $M$ be positive semidefinite.

1. Without putting any condition on the optimization problem, simply considering the minimization problem above, is it expected that one would recover a positive definite matrix?

2. If the question for $(1)$ is negative, are there recent results that show the positive definite completion and characterize the number of samples, successful recovery rate,...

Thank you for hints, directions or suggestions.