I am trying to implement NMF with Alternating Least Squares method. I am just curious about the following basic implementation of the problem: 

If I understand correctly, we can solve for each matrix equation stated in this pseudocode without nonnegativity constraints, with closed form solution and set the negative entries to 0, in a brute force way. Is this understanding correct? Is this a basic alternative to more complicated, constrained optimization problems, where we use projected gradient descent, for example? More importantly, if implemented in this basic way, will the algorithm have any practical value? I want to use NMF for variable reduction purposes and it is important that I use NMF, since my data is by definition non-negative. I am looking for opinions on this one. 

Answers: 

1. If I understand correctly, we can solve for each matrix equation stated in this pseudocode without nonnegativity constraints, with closed form solution and set the negative entries to 0, in a brute force way. Is this understanding correct? Yes.

2. Is this a basic alternative to more complicated, constrained optimization problems, where we use projected gradient descent, for example? ---In a sense, yes. This is indeed a fast way of Nonnegative factorization. However, articles related to NMF would point out that although this method is fast, it does not guarantee convergence of the nonnegative factors. A better implementation to use would be Hierarchical Alternating Least Squares for NMF (HALS-NMF). Check this paper for a comparison of some popular NMF algorithms: http://www.cc.gatech.edu/~hpark/papers/jgo.pdf

3. More importantly, if implemented in this basic way, will the algorithm have any practical value? Just basing from my experience, I would say that results aren't as good as compared to say HALS or BPP(Block Pivoting Principle).