Exact low rank matrix completion using nuclear norm minimization can be formulated as a semidefinite program (SDP). Following the notation in the paper, a convex problem for noisy matrix completion can be:

$$\text{minimize} \,\, \|X\|_* \quad \text{subject to} \quad \|\mathcal{P}_\Omega(X)-\mathcal{P}_\Omega(M)\|_2\le\epsilon \tag{*}$$

where $\mathcal{P}_\Omega(X)$ is the projection of $X$ onto the set of observed entries $\Omega$. This problem can be formulated as the SDP below SDP in $X, W_1, W_2$

$$\begin{array}{rl} \text{minimize} & \text{trace}(W_1)+\text{trace}(W_2)\\ \text{subject to} & \left[ \begin{array}{cc} \epsilon I & \mathcal{P}_\Omega(X-M) \\ \mathcal{P}_\Omega(X-M)^T & \epsilon I \end{array} \right] \succeq 0\\ & \left[ \begin{array}{cc} W_1 & X \\ X^T & W_2 \end{array} \right] \succeq 0\end{array}$$

The question is when the $2$-norm constraint of problem (*) is replaced by the Frobenius norm constraint, i.e.,

$$\text{minimize} \,\, \|X\|_* \quad \text{subject to} \quad \|\mathcal{P}_\Omega(X)-\mathcal{P}_\Omega(M)\|_F \le \epsilon$$

it can still be cast as a SDP as stated in here and many other papers. Could anyone tell me how to formulate this problem as a SDP?

Edit: follow up question.