Why is the matrix completion objective non-convex?

Question

Is there any general proof by taking second order convexity test to prove the following?

• Why does the non-negative matrix factorization problem non-convex?

I am self-learning convex optimization from a book, and it will be a great help if you guys can help me. Not a lot of reference are available on why the objective is non-convex here.

Answer 1

You should try asking more specific questions in which you show some of your own working if you would like to receive answers from this site. This is pretty late, but I am learning too and working on the same problem, so I'll give you the following suggestions for this problem:

• Prove/observe that the problem does not have one unique solution. For this, given a non-zero solution $X^*$, find another solution of the form $X^*C$. This will tell you your function is at least not strictly convex.
• Prove that two of your solutions from above do not form a convex set. (Two elements $x,y \in W$ form a convex set if $[xt + (1-t)y] \in W, \ t\in [0,1]$ ).

For more information you can see this related question I recently asked: Minimization of $||A-XX^T||^2_F$ with no restrictions

Happy learning!