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Let's say I have two sets of paired points $\{x_i\}_{i=1..n}$, $\{y_i\}_{i=1..n}$ defined in an $N$-dimensional space and I want to use the Kabsch algorithm to estimate the best rotation matrix / translation vector that transform each point in $\{y_i\}_{i=1..n}$ into its corresponding version in $\{x_i\}_{i=1..n}$ by solving the Orthogonal Procrustes problem based on the RMSD (root mean squared deviation) criterion:

\begin{equation} R = \arg \min_{R} \|R Y - X {\|}_F \end{equation} where:

  • $R$ is a rotation matrix (${R}^T R = I_N$),
  • $\| . {\|}_F$ is the Frobenius norm and $X$,
  • $Y$ is the matrix representation of $\{x_i\}_{i=1..n}$ and $\{y_i\}_{i=1..n}$.

Question: Knowing that the distribution of the random variable $x$ which generates each $x_i$ is multivariate Gaussian, is there any theoretical garentees about the distribution of the estimated vectors $\hat{y}_i$ using this algorithm ?

Attempt: I have doubts about this since the procedure is not Bayesian and there is no use of prior knowledge about the distribution of the target random variable $p(x)$. But I don't know whether this information is "implicitly" accounted for in the model.

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