Orthogonal Procrustes problem

Key takeaways

Orthogonal Procrustes problem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Orthogonal Procrustes problem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Orthogonal Procrustes problem from memory before moving on to harder problems.

Reference excerpt

The orthogonal Procrustes problem is a matrix approximation problem in linear algebra. In its classical form, one is given two matrices A {\displaystyle A} and B {\displaystyle B} and asked to find an orthogonal matrix Ω {\displaystyle \Omega } which most closely maps A {\displaystyle A} to B {\displaystyle B} . Specifically, the orthogonal Procrustes problem is an optimization problem given by

minimize Ω ‖ Ω A − B ‖ F subject to Ω T Ω = I , {\displaystyle {\begin{aligned}{\underset {\Omega }{\text{minimize}}}\quad &\|\Omega A-B\|_{F}\\{\text{subject to}}\quad &\Omega ^{T}\Omega =I,\end{aligned}}}

where ‖ ⋅ ‖ F {\displaystyle \|\cdot \|_{F}} denotes the Frobenius norm. This is a special case of Wahba's problem (with identical weights; instead of considering two matrices, in Wahba's problem the columns of the matrices are considered as individual vectors). Another difference is that Wahba's problem tries to find a proper rotation matrix instead of just an orthogonal one. The name Procrustes refers to a bandit from Greek mythology who made his victims fit his bed by either stretching their limbs or cutting them off.

Solution This problem was originally solved by Peter Schönemann in a 1964 thesis, and shortly after appeared in the journal Psychometrika. This problem is equivalent to finding the nearest orthogonal matrix to a given matrix M = B A T {\displaystyle M=BA^{T}} , i.e. solving the closest orthogonal approximation problem

min R ‖ R − M ‖ F s u b j e c t t o R T R = I {\displaystyle \min _{R}\|R-M\|_{F}\quad \mathrm {subject\ to} \quad R^{T}R=I} . To find matrix R {\displaystyle R} , one uses the singular value decomposition (for which the entries of Σ {\displaystyle \Sigma } are non-negative)

M = U Σ V T {\displaystyle M=U\Sigma V^{T}\,\!}

to write

R = U V T . {\displaystyle R=UV^{T}.\,\!}

Proof of Solution One proof depends on the basic properties of the Frobenius inner product that induces the Frobenius norm:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Orthogonal Procrustes problem

Start with the simplest possible case. Write down what Orthogonal Procrustes problem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Orthogonal Procrustes problem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Orthogonal Procrustes problem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Orthogonal Procrustes problem

In research: Orthogonal Procrustes problem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Orthogonal Procrustes problem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Orthogonal Procrustes problem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Linear algebra, Matrix theory, Singular value decomposition, so understanding it makes those chapters shorter.
In everyday life: Look for Orthogonal Procrustes problem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Orthogonal Procrustes problem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Orthogonal Procrustes problem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Orthogonal Procrustes problem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Orthogonal Procrustes problem in simple terms?

The orthogonal Procrustes problem is a matrix approximation problem in linear algebra. In its classical form, one is given two matrices A {\displaystyle A} and B {\displaystyle B} and asked to find an orthogonal matrix Ω {\displaystyle \Omega } which most closely maps A {\displaystyle A} to B {\dis…

Why does Orthogonal Procrustes problem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Orthogonal Procrustes problem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Orthogonal Procrustes problem.