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 $B$.
Questions tagged [procrustes-problem]
37 questions
26
votes
1 answer
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.,
$$\min_{Q^TQ=I_{n \times n}} \|A-Q\|_F$$
BadAtMath
- 439
14
votes
3 answers
Solve least-squares minimization from overdetermined 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:
$$
\mathop{\text{minimize }}_{X \in \mathbb{R}^{n \times k}} \left\| A X - B \right\|_F^2 \quad \text{ subject to } X^T X =…
Alec Jacobson
- 573
9
votes
1 answer
Find a permutation of the rows of a matrix that minimizes the sum of squared errors
I'm struggling with the following problem:
Let $A, B \in \mathbb R^{n \times d}$. Denote by $\mathcal{P}$ the set of all possible permutations of the rows of $A$. Find a permutation $\pi \in \mathcal{P}$ that minimizes $$\sum_{i=1}^n\sum_{j=1}^d…
Eva
- 111
5
votes
0 answers
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 $A$ and $B$ are $n\times d$. Geometrically, $M$…
Matt
- 51
5
votes
1 answer
Given $A_i, B_i \in \mathbb{R}^{k \times d}$, minimize $\sum_{i} \lVert U A_i V^T - B_i \rVert_F^2 $ over orthogonal $U, V$.
Given a collection of rectangular matrices $A_i, B_i \in \mathbb{R}^{k \times d}$ for $1 \leq i \leq n$, I am looking for an analytical solution for orthogonal matrices $U \in \mathbb{R}^{k \times k}$ and $V \in \mathbb{R}^{d \times d}$ with $U^T U…
tommym
- 483
4
votes
0 answers
Procrustes with inequality constraint
I am interested in a variant of the orthogonal Procrustes problem
$$ \begin{array}{ll} \underset{X} {\text{minimize} } & \| X A - B \|_{\text F}
\\ \text{subject to} & X^TX = I
\\ & (X^T C)_{ii} \geq D_{ii}
\end{array} $$
where…
xdaimon
- 107
3
votes
1 answer
Nearest Semi Orthogonal Matrix with Orthogonal Rows in the Frobenius Norm Sense (Projection onto Semi Orthogonal Matrices Set)
I found a question, Nearest Semi Orthogonal Matrix Using the Entry Wise ℓ1 Norm, about finding a nearest semi-orthogonal matrix, but I need to find the nearest semi-orthogonal matrix subject to a slightly different constraint. Given $m \times n$…
Buu Pham
- 77
3
votes
1 answer
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 norm is the Frobenius norm. The minimization here…
squattyroo
- 183
2
votes
0 answers
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.
Suppose $A$ is the identity matrix. I would like to…
Mael
- 793
2
votes
1 answer
Minimal solution to $\Vert A X B - C \rVert^2_F$ subject to $X^T X = I$
Given $A \in \mathbb{R}^{m \times n}$, $B \in \mathbb{R}^{n \times d}$ and $C \in \mathbb{R}^{m \times d}$, I want to find an analytical solution for $X \in \mathbb{R}^{n \times n}$ that minimizes
$$ \lVert A X B - C \rVert^2_F$$ subject to the…
tommym
- 483
2
votes
1 answer
Why projection onto the Stiefel manifold fails to solve the orthogonal Procrustes problem
The orthogonal Procrustes problem finds an orthogonal matrix $\Omega$ minimizing the Procrustes objective:
$$
\min_\Omega ||\Omega A - B||_F, \quad \Omega^\top \Omega = I
$$
It is well known that the solution is $\Omega^* = U V^\top$, where $U, V$…
calmcc
- 271
2
votes
1 answer
Solution to a Procrustes-like Problem
I recently came across the following problem that resembles a Procrustes problem and I wonder if an analytic solution for this problem might exist:
$$\underset{(R,\alpha)}{\operatorname{argmin}} ||RAe^{j\alpha W}-B||_F$$
Where $A,B \in \mathbb{C}^{3…
Mantabit
- 223
2
votes
1 answer
Determining a family of Solutions to an underconstrained Modified Wahba's Problem, $J = \sum_{i=1}^{N} \frac{1}{2} \|a_i - RQR^Tb_i \|^2$
I have a minimisation problem of the form:
$$
J(R) = \underset{R}{\mathrm{argmin}} \sum_{i=1}^{N} \frac{1}{2}\|a_i - RQR^T b_i \|^2
$$
Where $Q\in SO\{3\}$ and $a_i, b_i \in \mathbb{R}^3$ are known, and we wish to determine $R\in SO\{3\}$.
I am…
Damien
- 1,686
2
votes
1 answer
Optimizing Trace$(Q^TZ)$ subject to $Q^TQ=I$
Let $Z \in \mathbb{R}^{m \times n}$ be a tall matrix ($m > n$). Solve the following optimization problem in $Q \in \mathbb{R}^{m \times n}$
$$\begin{array}{ll} \text{maximize} & \mbox{Tr} \left(Q^T Z \right)\\ \text{subject to} & Q^T Q = I_{n…
kkcocoqq
- 316
2
votes
1 answer
Convexity of the orthogonal Procrustes problem
Given two orthogonal matrices ${\bf{M}} \in {{\Bbb{R}}^{m \times n}}$ and ${\bf{N}} \in {{\Bbb{R}}^{m \times n}}$, there is an orthogonal transformation matrix ${\bf{T}} \in {{\Bbb{R}}^{n \times n}}$ which closely maps ${\bf{M}}$ and…
ar_k
- 51