We discuss optimal low-rank approximation of matrices with non-negative entries, without the need of a regularization parameter. It will be shown that the standard SVD-approximation can be recovered via convex-optimization, which is why adding mild convex constraints often gives an optimal solution. Moreover, the issue of computability will be addressed by solving our new convex problem via the so-called Douglas-Rachford algorithm. We will see that if there is a unique optimal solution than also the non-convex Douglas-Rachford will locally converge to it.