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   (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(see Theorem 1.1 of Candes and Recht, 2008 [paper ][1] )

Let the underlying matrix $M$ be semi positive definite. 

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


  [1]: http://statweb.stanford.edu/~candes/papers/MatrixCompletion.pdf