I have an optimization problem of the form
$$ \text{minimize }_{\mathbf{U,P,Q}} \quad||\mathbf{X} - \mathbf{PQ}||^2 + ||\mathbf{Y-UP'}||^2 + ||\mathbf{U}||^2 $$
I can solve this using Alternating minimization (AM) i.e. in each iteration solve for one variable keeping the others constant and in subsequent iteration use the most recent value of the variable. Another approach could be, I can first solve for
$$ \text{minimize }_{\mathbf{P,Q}} \quad||\mathbf{X} - \mathbf{PQ}||^2 $$ using AM and then get values for $\mathbf{P,Q}$ and use the obtained value of $\mathbf{P}$ to solve
$$ \text{minimize }_{\mathbf{U}} \quad ||\mathbf{Y-UP'}||^2 + ||\mathbf{U}||^2 $$
exactly.
Is there any study on the quality of the two approaches ? What can we say about the approximation error in both cases ? Any pointers please ?