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Questions tagged [procrustes-problem]

The Orthogonal Procrustes Problem is a matrix approximation problem in linear algebra. In its classical form, one is given two matrices, $A$ and $B$, and is asked to find an orthogonal matrix $R$ that most closely maps $A$ to $B$.

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Procrustes Problem with Maximization (Instead of Minimization)

The classical (orthogonal) Procrustes problem is to solve the optimization problem $$ \begin{array}{rl} \min&\|\Omega{A}-B\|_F\\ \text{s.t.}&\Omega^\mathrm{T}\Omega=I \end{array} $$ The ...
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3D to 3D correspondence norm derivation

I've been going through a set of slides about a modified version of the Procrustes problem. The whole problem is described by trying to find a transformation that satisfies $$A_i = sRB_i + T$$ where $...
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Distance of matrix to $\mbox{SO}(n)$ w.r.t. Frobenius norm

Given $A \in \mathbb R^{n\times n}$, I was told that $$\mbox{dist}(A, \mbox{SO}(n)) = \inf_{Q \in \mbox{SO}(n)} |A-Q| \overset ? = \inf_{Q \in \mbox{SO}(n)} | Q^\top A - \text{Id} |$$ where we use ...
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Nearest signed permutation matrix to a given matrix $A$

Let $A \in \mathbb{R}^{n\times n}$ be a square matrix and let $Q \in O(n)$ be the nearest orthogonal matrix to $A$ under the Frobenius norm, i.e. $$Q = \text{arg}\min_{M \in O(n)} ||A - M||_{F}^2$$ ...
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Nearest (In the Frobenius Norm Sense) Semi Orthonormal Matrix - $ R {R}^{T} = I $

I found an entry about finding the nearest semi-orthonormal matrix: Nearest semi-orthonormal matrix using the entry-wise $1$-norm But I need to find the nearest semi-orthonormal matrix with a little ...
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Optimal Rotation between two sets of points [closed]

Let $p_i,q_j\in\mathbb{R}^3$ be sequences of points with $i,j\in\{1,2,3\}$. How can I solve the optimization problem $$D:=\arg\min_{R,t}\left(\max_{i=1,2,3}\Vert p_i-(Rq_i+t) \Vert\right),$$ where $...
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Least squares minimization subject to generalized orthogonality constraint?

I'd like to solve the following minimization problem: $$ \min_{\Phi^T D \Phi = I_k} \frac{1}{2}\|\Phi - Y\|_F^2, $$ where $\Phi \in \mathbb{R}^{n \times k}$, $D \in \mathbb{R}^{n \times n}$ is a (...
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1answer
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Deriving the Optimal Solution of the Orthogonal Procrustes Problem

I am trying to work through the Orthogonal Procrustes Problem but I do not understand a particular step. I would appreciate any help in understanding the steps the author goes from the first line to ...
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Solve Least Squares Minimization from Over Determined System with Orthonormal Constraint

I would like to find the rectangular matrix $X \in \mathbb{R}^{n \times k}$ that solves the following minimization problem: $$ \mathop{\text{minimize }}_{X \in \mathbb{R}^{n \times k}} \left\| A X - ...
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Showing that matrix $Q=UV^T$ is the nearest orthogonal matrix to $A$.

Let $A$ be an $m \times n$ matrix with a singular value decomposition $A=U\Sigma V^T$. Show that the matrix $Q=UV^T$ is the nearest orthogonal matrix to $A$, i.e., $$\min_{Q^TQ=I_{n \times n}} \|A-Q\...
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Find an orthogonal matrix to minimize the norm

Let $A\in \mathbb{R}^{n\times n}$. Find $\overline O$, orthogonal matrix, to minimize $\|A-O\|_F$. That is; $$\min_{O\in O(n)} \|A-O\|_F$$ Where $O(n)$ are the set of orthogonal matrices of $\...
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How to apply Procrustes transformation to a single point

I guess this is probably a too simple question for most of you but my algebra skills are very limited (ashamed). I have run two PCA analyses and conducted a Procrustes analyses to fit one matrix (...
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Orthogonal Procrustes Problem using the operator norm

If $A, B \in \mathbb{R}^{n \times r}$ are two matrices, the solution to the so-called Orthogonal Procrustes Problem $$\min_{O^TO=I_r} \|AO-B\|$$ is given by the polar factor of $A^TB$ whenever the ...
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Orthogonal Procrustes Problem

The classical Orthogonal Procrustes Problem is $$\begin{array}{ll} \text{minimize} & \|A\Omega-B\|_{F}\\ \text{subject to} & \Omega'\Omega=I\end{array}$$ where $A$ and $B$ are known matrices....
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Orthogonal Procrustes Variant

(author note: this question was also asked on mathoverflow). The orthogonal Procrustes problem seeks a matrix $M$ that minimizes $||AM-B||_F$ subject to $M^TM=I$, where $M$ is $d\times d$ and both $A$...
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Orthogonal procrustes problem using quaternions

Hello I'm trying solve orthogonal procrustes problem in 3d with a help of quaternions. Original problem is: For matrix $A$ find orthogonal matrix $Q$ that $$||A-Q||_F =\min_{\Omega \in SO(3)} ||A-\...
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Variant of orthogonal Procrustes problem

I want to find an anti-symmetric matrix $T$ which minimizes $\|A-e^TBe^{-T}\|^2 + \mu\|T\|^2$, where $A$ and $B$ are symmetric positive definite matrices and the norm is the Frobenius matrix norm. The ...