The closest matrix to a given matrix in $\text{SO}(n)$ is unique iff the smallest singular value is a strict minimum

linear algebramatrix-calculusoptimizationorthogonal matricesreference-request

$\newcommand{\SOn}{\operatorname{SO}_n}$
$\newcommand{\On}{\operatorname{O}_n}$
$\newcommand{\Sym}{\operatorname{Sym}_n}$
$\newcommand{\Skew}{\operatorname{Skew}_n}$
$\newcommand{\dist}{\operatorname{dist}}$
$\newcommand{\Sig}{\Sigma}$
$\newcommand{\sig}{\sigma}$
$\newcommand{\al}{\alpha}$
$\newcommand{\id}{\operatorname{Id}}$

This question is a reference request.

Claim:

Let $A $ be a $n \times n$ real matrix with non-positive determinant.

Then there is a unique closest matrix $Q \in \SOn$ to $A$ (w.r.t the Euclidean Frobenius norm) if and only if the smallest singular value of $A$ is strictly smaller than the rest of the singular values.

Question: Is this claim "known"? Where can I find a reference for it?

Note that I am looking for a reference, not a proof. (I have a proof…)

Also, I am specifically asking for the distance minimizer in $\SOn$. If we replace $\SOn$ with $\On$, then the minimizer(s) is the orthogonal polar factor from polar decomposition. (and is unique whenever the matrix $A$ is invertible).

Best Answer

A casual google search reveals that the result was mentioned in Myronenko and Song (2009), On the closed-form solution of the rotation matrix arising in computer vision problems, arXiv:0904.1613v1:

Lemma 1. Let $R_{D\times D}$ be an unknown rotation matrix and $A_{D\times D}$ be a known real square matrix. Let $USV^T$ be a Singular Value Decomposition (SVD) of $A$, where $UU^T=VV^T=I,\ S=d(s_i),\ s_1\ge s_2\ge\cdots\ge s_D\ge0$. Then the optimal rotation matrix $R$ that maximizes $\operatorname{tr}(A^TR)$ is $$ R=UCV^T,\ \text{ where } C=d(1,1,\ldots,1,\det(UV^T)). $$ Matrix $R$ is unique for any $A$, except for two cases:

  1. $\operatorname{rank}(A)<D-1$,
  2. $\det(A)<0$ and the smallest singular value, $s_D$, is not distinct.

However, as orthogonal Procrustes problem and its variants have almost been studied to death, I am certain that the result had been mentioned by other earlier papers.

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