Polar factorization theorem

Key takeaways

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

Reference excerpt

In optimal transport, a branch of mathematics, polar factorization of vector fields is a basic result due to Brenier (1987), with antecedents of Knott-Smith (1984) and Rachev (1985), that generalizes many existing results among which are the polar decomposition of real matrices, and the rearrangement of real-valued functions.

The theorem Notation. Denote ξ # μ {\displaystyle \xi _{\#}\mu } the image measure of μ {\displaystyle \mu } through the map ξ {\displaystyle \xi } . Definition: Measure preserving map. Let ( X , μ ) {\displaystyle (X,\mu )} and ( Y , ν ) {\displaystyle (Y,\nu )} be some probability spaces and σ : X → Y {\displaystyle \sigma :X\rightarrow Y} a measurable map. Then, σ {\displaystyle \sigma } is said to be measure preserving iff σ # μ = ν {\displaystyle \sigma _{\#}\mu =\nu } , where # {\displaystyle \#} is the pushforward measure. Spelled out: for every ν {\displaystyle \nu } -measurable subset Ω {\displaystyle \Omega } of Y {\displaystyle Y} , σ − 1 ( Ω ) {\displaystyle \sigma ^{-1}(\Omega )} is μ {\displaystyle \mu } -measurable, and μ ( σ − 1 ( Ω ) ) = ν ( Ω ) {\displaystyle \mu (\sigma ^{-1}(\Omega ))=\nu (\Omega )} . The latter is equivalent to:

∫ X ( f ∘ σ ) ( x ) μ ( d x ) = ∫ X ( σ ∗ f ) ( x ) μ ( d x ) = ∫ Y f ( y ) ( σ # μ ) ( d y ) = ∫ Y f ( y ) ν ( d y ) {\displaystyle \int _{X}(f\circ \sigma )(x)\mu (dx)=\int _{X}(\sigma ^{*}f)(x)\mu (dx)=\int _{Y}f(y)(\sigma _{\#}\mu )(dy)=\int _{Y}f(y)\nu (dy)}

where f {\displaystyle f} is ν {\displaystyle \nu } -integrable and f ∘ σ {\displaystyle f\circ \sigma } is μ {\displaystyle \mu } -integrable. Theorem. Consider a map ξ : Ω → R d {\displaystyle \xi :\Omega \rightarrow \mathbb {R} ^{d}} where Ω {\displaystyle \Omega } is a convex subset of R d {\displaystyle \mathbb {R} ^{d}} , and μ {\displaystyle \mu } a measure on Ω {\displaystyle \Omega } which is absolutely continuous. Assume that ξ # μ {\displaystyle \xi _{\#}\mu } is absolutely continuous. Then there is a convex function φ : Ω → R {\displaystyle \varphi :\Omega \rightarrow \mathbb {R} } and a map σ : Ω → Ω {\displaystyle \sigma :\Omega \rightarrow \Omega } preserving μ {\displaystyle \mu } such that

ξ = ( ∇ φ ) ∘ σ {\displaystyle \xi =\left(\nabla \varphi \right)\circ \sigma }

In addition, ∇ φ {\displaystyle \nabla \varphi } and σ {\displaystyle \sigma } are uniquely defined almost everywhere.

Applications and connections

Dimension 1 In dimension 1, and when μ {\displaystyle \mu } is the Lebesgue measure over the unit interval, the result specializes to Ryff's theorem. When d = 1 {\displaystyle d=1} and μ {\displaystyle \mu } is the uniform distribution over [ 0 , 1 ] {\displaystyle \left[0,1\right]} , the polar decomposition boils down to

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Polar factorization theorem

Start with the simplest possible case. Write down what Polar factorization theorem 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 Polar factorization theorem 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 Polar factorization theorem 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 Polar factorization theorem

In research: Polar factorization theorem 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 Polar factorization theorem 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: Polar factorization theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Measures (measure theory), Theorems involving convexity, so understanding it makes those chapters shorter.
In everyday life: Look for Polar factorization theorem 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 Polar factorization theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Polar factorization theorem 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 Polar factorization theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Polar factorization theorem in simple terms?

In optimal transport, a branch of mathematics, polar factorization of vector fields is a basic result due to Brenier (1987), with antecedents of Knott-Smith (1984) and Rachev (1985), that generalizes many existing results among which are the polar decomposition of real matrices, and the rearrangeme…

Why does Polar factorization theorem 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 Polar factorization theorem?

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 Polar factorization theorem.