I am trying to implement the algorithm for matrix completion proposed by Keshavan, Montanari and Oh (2009). It consists of three steps:
Trimming which nulls some rows and columns to make the high singular values stand out
Projecting which basically does a SVD of a 'sparse' matrix
Cleaning which minimizes the residual errors
I have some problems understanding the cleaning step: Projecting gives you $U$, $\Sigma$ and $V$ (eq 3). Cleaning requires a $S$ matrix in $R(r,r)$ minimizing the cost function (eq 4). Optimizing the cost function with the Gradient Descent algorithm (below Remark 6.2) requires $S$. However, I cannot find any information how to calculate $S$. Can anybody please bring some light into how to understand these equations?