Hi,  

I am optimizing a function over a matrix $U$, where $U \in \mathbb{R}^{m \times n}$ and $U^TU = I$. I do not want to run a constrained maximization program, since employing the constraint $U^TU = 1$ with lagrange multipliers would lead to higher time for convergence. I instead want to run simple gradient based optimization. So, I wanted to encode the orthogonality constraint into the input variable itself. What I mean by that is, for example, if
$U$ was a square matrix then by Cayley transform for a skew-symmetric matrix $A$ we have:
$U = (I-A)(I+A)^{-1}$ -- and then I can use unconstrained optimization with a lower dimensionality of input variables.


So my question is whether there is an extension of the Cayley transform to non-square matrices of type $U^TU = I$ where $U \in \mathbb{R}^{m \times n}$.


Thanks  

I.J



cayley transform for non-square matrices
Have a look at the following slides (several pointers are in there)


[Optimization on the Stiefel manifold](http://www.inma.ucl.ac.be/~absil/Talks/GAMM_Zurich_2009-09-10_Stiefel_03.pdf)


The point is that you can directly remain on the manifold while optimizing, so no explicit "constraint enforcement" will be required.


