Polar factorization theorem

Notation. Denote $\backslash xi\_\backslash \#\; \backslash mu$ the image measure of $\backslash mu$ through the map $\backslash xi$.

Definition: Measure preserving map. Let $(X,\backslash mu)$ and $(Y,\backslash nu)$ be some probability spaces and $\backslash sigma\; :X\; \backslash rightarrow\; Y$ a measurable map. Then, $\backslash sigma$ is said to be measure preserving iff $\backslash sigma\_\{\backslash \#\}\backslash mu\; =\; \backslash nu$, where $\backslash \#$ is the pushforward measure. Spelled out: for every $\backslash nu$-measurable subset $\backslash Omega$ of $Y$, $\backslash sigma^\{-1\}(\backslash Omega)$ is $\backslash mu$-measurable, and $\backslash mu(\backslash sigma^\{-1\}(\backslash Omega))=\backslash nu(\backslash Omega\; )$. The latter is equivalent to:

:$\backslash int\_\{X\}(f\backslash circ\; \backslash sigma)(x)\; \backslash mu(dx)\; =\backslash int\_X\; (\backslash sigma^*f)(x)\; \backslash mu(dx)\; =\backslash int\_Y\; f(y)\; (\backslash sigma\_\{\backslash \#\}\backslash mu)(dy)\; =\; \backslash int\_\{Y\}f(y)\; \backslash nu(dy)$

where $f$ is $\backslash nu$-integrable and $f\backslash circ\; \backslash sigma$ is $\backslash mu$-integrable.

Theorem. Consider a map $\backslash xi\; :\backslash Omega\; \backslash rightarrow\; R^\{d\}$ where $\backslash Omega$ is a convex subset of $R^\{d\}$, and $\backslash mu$ a measure on $\backslash Omega$ which is absolutely continuous. Assume that $\backslash xi\_\{\backslash \#\}\backslash mu$ is absolutely continuous. Then there is a convex function $\backslash varphi\; :\backslash Omega\; \backslash rightarrow\; R$ and a map $\backslash sigma\; :\backslash Omega\; \backslash rightarrow\; \backslash Omega$ preserving $\backslash mu$ such that

$\backslash xi\; =\backslash left(\; \backslash nabla\; \backslash varphi\; \backslash right)\; \backslash circ\; \backslash sigma$

In addition, $\backslash nabla\; \backslash varphi$ and $\backslash sigma$ are uniquely defined almost everywhere.{{cite book |last1=Santambrogio |first1=Filippo |title=Optimal transport for applied mathematicians |date=2015 |publisher=Birkäuser |location=New York |citeseerx=10.1.1.726.35 }}

In dimension 1, and when $\backslash mu$ is the Lebesgue measure over the unit interval, the result specializes to Ryff's theorem.{{cite journal |last1=Ryff |first1=John V. |title=Orbits of L1-Functions Under Doubly Stochastic Transformation |journal=Transactions of the American Mathematical Society |date=1965 |volume=117 |pages=92–100 |doi=10.2307/1994198 |jstor=1994198 |url=https://www.jstor.org/stable/1994198 |access-date=16 April 2021|url-access=subscription }} When $d=1$ and $\backslash mu$ is the uniform distribution over $\backslash left[0,1\backslash right]$, the polar decomposition boils down to

$\backslash xi\; \backslash left(\; t\backslash right)\; =F\_\{X\}^\{-1\}\backslash left(\; \backslash sigma\; \backslash left(\; t\backslash right)\; \backslash right)$

where $F\_\{X\}$ is cumulative distribution function of the random variable $\backslash xi\; \backslash left(\; U\backslash right)$ and $U$ has a uniform distribution over $\backslash left[\; 0,1\backslash right]$. $F\_\{X\}$ is assumed to be continuous, and $\backslash sigma\; \backslash left(\; t\backslash right)=F\_\{X\}\backslash left(\; \backslash xi\; \backslash left(\; t\backslash right)\; \backslash right)$ preserves the Lebesgue measure on $\backslash left[\; 0,1\backslash right]$.

When $\backslash xi$ is a linear map and $\backslash mu$ is the Gaussian normal distribution, the result coincides with the polar decomposition of matrices. Assuming $\backslash xi\; \backslash left(\; x\backslash right)\; =Mx$ where $M$ is an invertible $d\backslash times\; d$ matrix and considering $\backslash mu$ the $\backslash mathcal\{N\}\backslash left(\; 0,I\_\{d\}\backslash right)$ probability measure, the polar decomposition boils down to

M=SO

where $S$ is a symmetric positive definite matrix, and $O$ an orthogonal matrix. The connection with the polar factorization is $\backslash varphi\; \backslash left(x\backslash right)\; =x^\{\backslash top\; \}Sx/2$ which is convex, and $\backslash sigma\; \backslash left(\; x\backslash right)\; =Ox$ which preserves the $\backslash mathcal\{N\}\backslash left(\; 0,I\_\{d\}\backslash right)$ measure.

The results also allow to recover Helmholtz decomposition. Letting $x\backslash rightarrow\; V\backslash left(\; x\backslash right)$ be a smooth vector field it can then be written in a unique way as

V=w+\nabla p

where $p$ is a smooth real function defined on $\backslash Omega$, unique up to an additive constant, and $w$ is a smooth divergence free vector field, parallel to the boundary of $\backslash Omega$.

The connection can be seen by assuming $\backslash mu$ is the Lebesgue measure on a compact set $\backslash Omega\; \backslash subset\; R^\{n\}$ and by writing $\backslash xi$ as a perturbation of the identity map

\xi _{\epsilon }(x)=x+\epsilon V(x)

where $\backslash epsilon$ is small. The polar decomposition of $\backslash xi\; \_\{\backslash epsilon\; \}$ is given by $\backslash xi\; \_\{\backslash epsilon\; \}=(\backslash nabla\; \backslash varphi\_\{\backslash epsilon\; \})\backslash circ\; \backslash sigma\_\{\backslash epsilon\; \}$. Then, for any test function $f:R^\{n\}\backslash rightarrow\; R$ the following holds:

\int_{\Omega }f(x+\epsilon V(x))dx=\int_{\Omega }f((\nabla \varphi

_{\epsilon })\circ \sigma _{\epsilon }\left( x\right) )dx=\int_{\Omega

}f(\nabla \varphi _{\epsilon }\left( x\right) )dx

where the fact that $\backslash sigma\; \_\{\backslash epsilon\; \}$ was preserving the Lebesgue measure was used in the second equality.

In fact, as $\backslash textstyle\; \backslash varphi\; \_\{0\}(x)=\backslash frac\{1\}\{2\}\backslash Vert\; x\backslash Vert\; ^\{2\}$, one can expand $\backslash textstyle\; \backslash varphi\; \_\{\backslash epsilon\; \}(x)=\backslash frac\{1\}\{2\}\backslash Vert\; x\backslash Vert\; ^\{2\}+\backslash epsilon\; p(x)+O(\backslash epsilon\; ^\{2\})$, and therefore $\backslash textstyle\; \backslash nabla\; \backslash varphi\_\{\backslash epsilon\; \}\backslash left(\; x\backslash right)\; =x+\backslash epsilon\; \backslash nabla\; p(x)+O(\backslash epsilon\; ^\{2\})$. As a result, $\backslash textstyle\; \backslash int\_\{\backslash Omega\; \}\backslash left(\; V(x)-\backslash nabla\; p(x)\backslash right)\; \backslash nabla\; f(x))dx$ for any smooth function $f$, which implies that $w\backslash left(\; x\backslash right)\; =V(x)-\backslash nabla\; p(x)$ is divergence-free.{{cite book |last1=Villani |first1=Cédric |title=Topics in optimal transportation |date=2003 |publisher=American Mathematical Society}}

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Category:Theorems involving convexity