Misplaced Pages

Polar factorization theorem

Article snapshot taken from[REDACTED] with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
Theorem in Optimal Transport

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 R^{d}} where Ω {\displaystyle \Omega } is a convex subset of R d {\displaystyle 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 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} , the polar decomposition boils down to

ξ ( t ) = F X 1 ( σ ( t ) ) {\displaystyle \xi \left(t\right)=F_{X}^{-1}\left(\sigma \left(t\right)\right)}

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

Polar decomposition of matrices

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

M = S O {\displaystyle M=SO}

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

Helmholtz decomposition

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

V = w + p {\displaystyle V=w+\nabla p}

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

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

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

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

Ω f ( x + ϵ V ( x ) ) d x = Ω f ( ( φ ϵ ) σ ϵ ( x ) ) d x = Ω f ( φ ϵ ( x ) ) d x {\displaystyle \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 σ ϵ {\displaystyle \sigma _{\epsilon }} was preserving the Lebesgue measure was used in the second equality.

In fact, as φ 0 ( x ) = 1 2 x 2 {\displaystyle \textstyle \varphi _{0}(x)={\frac {1}{2}}\Vert x\Vert ^{2}} , one can expand φ ϵ ( x ) = 1 2 x 2 + ϵ p ( x ) + O ( ϵ 2 ) {\displaystyle \textstyle \varphi _{\epsilon }(x)={\frac {1}{2}}\Vert x\Vert ^{2}+\epsilon p(x)+O(\epsilon ^{2})} , and therefore φ ϵ ( x ) = x + ϵ p ( x ) + O ( ϵ 2 ) {\displaystyle \textstyle \nabla \varphi _{\epsilon }\left(x\right)=x+\epsilon \nabla p(x)+O(\epsilon ^{2})} . As a result, Ω ( V ( x ) p ( x ) ) f ( x ) ) d x {\displaystyle \textstyle \int _{\Omega }\left(V(x)-\nabla p(x)\right)\nabla f(x))dx} for any smooth function f {\displaystyle f} , which implies that w ( x ) = V ( x ) p ( x ) {\displaystyle w\left(x\right)=V(x)-\nabla p(x)} is divergence-free.

See also

  • polar decomposition – Representation of invertible matrices as unitary operator multiplying a Hermitian operator

References

  1. ^ Brenier, Yann (1991). "Polar factorization and monotone rearrangement of vector‐valued functions" (PDF). Communications on Pure and Applied Mathematics. 44 (4): 375–417. doi:10.1002/cpa.3160440402. Retrieved 16 April 2021.
  2. Knott, M.; Smith, C. S. (1984). "On the optimal mapping of distributions". Journal of Optimization Theory and Applications. 43: 39–49. doi:10.1007/BF00934745. S2CID 120208956. Retrieved 16 April 2021.
  3. Rachev, Svetlozar T. (1985). "The Monge–Kantorovich mass transference problem and its stochastic applications" (PDF). Theory of Probability & Its Applications. 29 (4): 647–676. doi:10.1137/1129093. Retrieved 16 April 2021.
  4. Santambrogio, Filippo (2015). Optimal transport for applied mathematicians. New York: Birkäuser. CiteSeerX 10.1.1.726.35.
  5. Ryff, John V. (1965). "Orbits of L1-Functions Under Doubly Stochastic Transformation". Transactions of the American Mathematical Society. 117: 92–100. doi:10.2307/1994198. JSTOR 1994198. Retrieved 16 April 2021.
  6. Villani, Cédric (2003). Topics in optimal transportation. American Mathematical Society.
Convex analysis and variational analysis
Basic concepts
Topics (list)
Maps
Main results (list)
Sets
Series
Duality
Applications and related
Measure theory
Basic concepts
Sets
Types of measures
Particular measures
Maps
Main results
Other results
For Lebesgue measure
Applications & related
Categories:
Polar factorization theorem Add topic