In nuclear norm based matrix completion, there exists a low rank matrix $M$ of which only the indicies in $\Omega$ are sampled. Candes and Recht (2008) recover the original matrix using following optimization problem.
$$ \begin{align} \text{minimize } &\quad \| X \|_* \tag{P1}\\ \text{subject to } &\quad X_{ij} = M_{ij}\quad (i,j) \in \Omega. \end{align} $$
Then, they prove that $M$ is actually the minimum of P1 by showing that there exists a "dual certificate" $\lambda$ such that
$$ P_\Omega(\lambda) \in \partial \| M \|_* $$
where $P_\Omega$ is the sampling operator that sets all elements not in $\Omega$ to zero, and $\partial \| M\|_*$ denotes the subdifferential of the nuclear norm at $M$.
I am trying to understand the intuition behind this particular dual certificate. For a given, $M$, can I construct that dual certificate and gain some understanding about this solution? For example, there is this idea of "shadow prices," where the Lagrangian variable (which is just the dual certificate), is somehow the change in the cost if you "relax" the constraint. Can I somehow similarly interpret the dual certificate here?
For example, would it be correct to say that the $(i,j)$th element of $\lambda$ is the amount that the nuclear norm of the solution would change if I increased the value of $M_{ij}$, assuming $(i,j) \in \Omega$?
