I'm trying to prove the convergence of Matrix factorization.
The problem is described below.
$|X-WH|^2 + |H|\_2^2 +|W|\_2^2$.


My optimization steps are using Alternating least squares which update H with fixing W and update W with fixing H. 


Although I can prove the convergence of subproblems(e.x. update H with fixing W), I have no idea to prove the convergence of this problem but not subproblems.


Does someone give some clues about how to prove the convergence of matrix factorization globally?


Thanks. 



The convergence of Matrix factorization
Your problem is a special case of **Alternating Minimization**, on which many papers have been written.


One of the clearest (for your nonconvex problem) is this classic paper, which pays particular attention to the two block case: L. Grippo, M. Sciandrone, *On the convergence of the block nonlinear Gauss–Seidel method under convex constraints*. Operation Research Letters 26(2000), 127--136.


