What is the best low-rank approximation of a non-negative matrix?

Question

I have a non-negative matrix $X$ and I compute its Singular Value Decomposition: $$X = U \Sigma V^T$$ then, I take the lower rank approximization: $$X_k = U_k \Sigma_k V^T_k$$ where $k < rank(X)$, $U_k$ is made of the first $k$ columns of $U$, $\Sigma_k$ is the $k \times k$ diagonal matrix whose elements are the first $k$ singualar values of $X$ and $V_k$ is composed of the first $k$ columns of the $V$ matrix.

$$X = X_k + N_{svd}$$

My measure of "loss of information" is $||X-X_k||_F^2$.

My question is:

1. Can I minimize this loss any further?

2. What is the idea of non-negative matrix factorization under noisy separability. and is it an answer to Q1?