I am trying to find a unitary tramsformation, $M$, that minimizes $\Vert MA-B \Vert_F^2$ where $A$ and $B$ are $N\times L,\;L\ge N$.
I know how to solve it without the unitary constraint. I thought using Lagrange multipliers with the constraint $\Vert M^HM-I \Vert_F^2 = tr\left\{ \left( M^HM-I \right)^H \left( M^HM-I \right)\right\} = 0$ but the it is quite difficult to solve. Is there any simpler way?
Thanks.
Best Answer
Since $M$ is unitary you can write $$\Vert MA-B\Vert_F^2=\Vert MA\Vert_F^2-2\langle MA,B\rangle_F+\Vert B\Vert_F^2\\ = \Vert A\Vert_F^2-2\mathrm{Re}\langle M,BA^*\rangle_F+\Vert B\Vert_F^2.$$ Therefore, your optimization reduces to maximizing $\mathrm{Re}\langle M,BA^*\rangle_F$. Using H$\ddot{\text{o}}$lder inequality and SVD of $BA^*=USV^*$, it's straightforward to show that the maximum is the Schatten 1-norm of $BA^*$ which is attained at $M=UV^*$.