Polar factorization theorem explained

In optimal transport, a branch of mathematics, polar factorization of vector fields is a basic result due to Brenier (1987),^[1] with antecedents of Knott-Smith (1984)^[2] and Rachev (1985),^[3] 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 → Y a measurable map. Then, \sigma is said to be measure preserving iff \sigma _{\#} \mu = \nu , where \# is the pushforward measure. Spelled out: for every \nu -measurable subset \Omega of Y , \sigma ^{-1} (\Omega) is \mu -measurable, and \mu(\sigma ^{-1} (\Omega)) = \nu(\Omega ) . The latter is equivalent to: \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 is \nu -integrable and f\circ \sigma is \mu -integrable.

Theorem. Consider a map

\xi :\Omega → R ^{d} where \Omega is a convex subset of R ^{d} , 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 → R and a map \sigma :\Omega → \Omega preserving \mu such that\xi = \left( \nabla \varphi \right) \circ \sigma

In addition,

\nabla \varphi and \sigma are uniquely defined almost everywhere.^[4]

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.^[5] When d =1 and \mu is the uniform distribution over \left[ 0 , 1 \right] , the polar decomposition boils down to\xi \left( t\right) = F ^{-1} _{X} \left( \sigma \left( t\right) \right)

where

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

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 x d matrix and considering \mu the l{N}\left( 0 ,I _{d} \right) probability measure, the polar decomposition boils down toM = 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 ^{\top} Sx/ 2 which is convex, and \sigma \left( x\right) = Ox which preserves the l{N}\left( 0 ,I _{d} \right) measure.

Helmholtz decomposition

The results also allow to recover Helmholtz decomposition. Letting

x → V\left( x\right) be a smooth vector field it can then be written in a unique way asV = 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 ^{n} and by writing \xi as a perturbation of the identity map\xi _{\epsilon} (x) = x + \epsilon V(x)

where

\epsilon is small. The polar decomposition of \xi _{\epsilon} is given by \xi _{\epsilon} = (\nabla \varphi _{\epsilon} )\circ \sigma _{\epsilon} . Then, for any test function f:R ^{n} → 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

\sigma _{\epsilon} was preserving the Lebesgue measure was used in the second equality.

In fact, as

style \varphi _{0} (x) = (1)/(2) \Vert x\Vert ^{2} , one can expand style \varphi _{\epsilon} (x) = (1)/(2) \Vert x\Vert ^{2} + \epsilon p(x) + O(\epsilon ^{2} ) , and therefore style \nabla \varphi _{\epsilon} \left( x\right) = x + \epsilon \nabla p(x) + O(\epsilon ^{2} ) . As a result, style \int _{\Omega} \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.^[6]

Notes and References

1. Brenier . Yann . Polar factorization and monotone rearrangement of vector‐valued functions . Communications on Pure and Applied Mathematics . 1991 . 44 . 4 . 375–417 . 10.1002/cpa.3160440402 . 16 April 2021.
2. Knott . M. . Smith . C. S. . On the optimal mapping of distributions . Journal of Optimization Theory and Applications . 1984 . 43 . 39–49 . 10.1007/BF00934745 . 120208956 . 16 April 2021.
3. Rachev . Svetlozar T. . The Monge–Kantorovich mass transference problem and its stochastic applications . Theory of Probability & Its Applications . 1985 . 29 . 4 . 647–676 . 10.1137/1129093 . 16 April 2021.
4. Book: Santambrogio . Filippo . Optimal transport for applied mathematicians . 2015 . Birkäuser . New York . 10.1.1.726.35 .
5. Ryff . John V. . Orbits of L1-Functions Under Doubly Stochastic Transformation . Transactions of the American Mathematical Society . 1965 . 117 . 92–100 . 10.2307/1994198 . 1994198 . 16 April 2021.
6. Book: Villani . Cédric . Topics in optimal transportation . 2003 . American Mathematical Society.