Polar Factorization Theorem

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 $\xi_\# \mu$ the image measure of $\mu$ through the map $\xi$.

'' Definition: Measure preserving map. '' Let $(X,\mu)$ and $(Y,\nu)$ be some probability spaces and $\sigma :X \rightarrow Y$ a map. Then, $\sigma$ is said to be measure preserving if for every $\nu$-measurable subset $\Omega$ of $Y$, $\sigma^(\Omega)$ is $\mu$-measurable and $\mu(\sigma^(\Omega))=\nu(\Omega )$, that is: $\textstyle \int_f\circ \sigma d\mu =\int_fd\nu$ with $f$ that is $\nu$-integrable and $\sigma \circ f$ that is $\mu$-integrable.

'' Theorem. '' Consider a map $\xi :\Omega \rightarrow R^$ where $\Omega$ is a convex subset of $R^$, and $\mu$ a measure on $\Omega$ which is absolutely continuous. Assume that $\xi_\mu$ is absolutely continuous. Then there is a convex function $\varphi :\Omega \rightarrow R$ and a map $\sigma :\Omega \rightarrow \Omega$ preserving $\mu$ such that

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

In addition, $\nabla \varphi$ and $\sigma$ are uniquely defined almost everywhere.

Applications and connections

Dimension 1

In dimension 1, and when $\mu$ is the Lebesgue measure over the unit interval, the result specializes to Ryff's theorem. When $d=1$ and $\mu$ is the uniform distribution over \left ,1\right/math>, the polar decomposition boils down to

$\xi \left( t\right) =F_^\left( \sigma \left( t\right) \right)$

where $F_$ is cumulative distribution function of the random variable $\xi \left( U\right)$ and $U$ has a uniform distribution over \left 0,1\right/math>. $F_$ is assumed to be continuous, and $\sigma \left( t\right)=F_\left( \xi \left( t\right) \right)$ preserves the Lebesgue measure on \left 0,1\right/math>.

Polar decomposition of matrices

When $\xi$ is a linear map and $\mu$ is the Gaussian normal distribution, the result coincides with the polar decomposition of matrices. Assuming $\xi \left( x\right) =Mx$ where $M$ is an invertible $d\times d$ matrix and considering $\mu$ the $\mathcal\left( 0,I_\right)$ 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 $\varphi \left(x\right) =x^Sx/2$ which is convex, and $\sigma \left( x\right) =Ox$ which preserves the $\mathcal\left( 0,I_\right)$ measure.

Helmholtz decomposition

The results also allow to recover Helmholtz decomposition. Letting $x\rightarrow V\left( x\right)$ 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 $\Omega$, unique up to an additive constant, and $w$ is a smooth divergence free vector field, parallel to the boundary of $\Omega$.

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

$\xi _(x)=x+\epsilon V(x)$

where $\epsilon$ is small. The polar decomposition of $\xi _$ is given by $\xi _=(\nabla \varphi_)\circ \sigma_$. Then, for any test function $f:R^\rightarrow R$ the following holds:

$\int_f(x+\epsilon V(x))dx=\int_f((\nabla \varphi _)\circ \sigma _\left( x\right) )dx=\int_f(\nabla \varphi _\left( x\right) )dx$

where the fact that $\sigma _$ was preserving the Lebesgue measure was used in the second equality.

In fact, as $\textstyle \varphi _(x)=\frac\Vert x\Vert ^$, one can expand $\textstyle \varphi _(x)=\frac\Vert x\Vert ^+\epsilon p(x)+O(\epsilon ^)$, and therefore $\textstyle \nabla \varphi_\left( x\right) =x+\epsilon \nabla p(x)+O(\epsilon ^)$. As a result, $\textstyle \int_\left( V(x)-\nabla p(x)\right) \nabla f(x))dx$ for any smooth function $f$, which implies that $w\left( x\right) =V(x)-\nabla p(x)$ is divergence-free.

See also

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References

{{Measure theory

Measures (measure theory)
Theorems involving convexity