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I have seen somewhere doing the following "trick" to simplify an optimization problem. I would like to understand the logic behind it.

Take two matrices $A$ and $B$ of dimension $N \times N$ and suppose that matrix $B$ is function of a parameter $\theta$, so we have $B(\theta)$. I have seen doing the following:

\begin{align} \min_{\theta} \|A - B(\theta)B(\theta)^\top\|_F^2 \rightarrow \min_{\theta, U} \|A^{1/2} - B(\theta)U\|_F^2 \end{align}

where $A^{1/2}$ is a square root of $A$ and $U$ is an orthogonal matrix, i.e. $UU^\top=U^\top U=I$. This would lead to an optimization problem which is different from the initial one but easier in the variable $\theta$, since reduces its order (for instance from quadratic to linear), at the expenses of also estimating the matrix $U$. Computing $U$ can be done through the procrustes method. Sometimes such a problem is said to be "over the Stiefel manifold".

Can you elucidate me how the two problems are related? Thanks.

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The stiefel manifolds is the matrix manifolds associated with orthogonal matrixes(i.e. verify the condition that you stated), as for the procrustes method it is just a generalization of least squares problem, I suggest that you see "Multivariate Data Analysis on Matrix Manifolds" for illustration. Naturally, approximating B(θ) with the matrix U will itself induces a subproblem,in the hope that the overall computational cost/feasability is advantagaeous when compared to the original problem. Finally, Regarding your inqury about the preservation of the minimizer, it should be obvious that the projective space contain it, otherwise it should be proven. Kindly,

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  • $\begingroup$ Hi Abidine, than you for your answer. Can you explain a bit more about the projective space? I also updated and revised the answer. $\endgroup$
    – yes
    Apr 19, 2022 at 15:07

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