Polar factorization theorem

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

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Notation. Denote   the image measure of   through the map  .

Definition: Measure preserving map. Let   and   be some probability spaces and   a measurable map. Then,   is said to be measure preserving iff  , where   is the pushforward measure. Spelled out: for every  -measurable subset   of  ,   is  -measurable, and  . The latter is equivalent to:

 

where   is  -integrable and   is  -integrable.

Theorem. Consider a map   where   is a convex subset of  , and   a measure on   which is absolutely continuous. Assume that   is absolutely continuous. Then there is a convex function   and a map   preserving   such that

 

In addition,   and   are uniquely defined almost everywhere.[1][4]

Applications and connections

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Dimension 1

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In dimension 1, and when   is the Lebesgue measure over the unit interval, the result specializes to Ryff's theorem.[5] When   and   is the uniform distribution over  , the polar decomposition boils down to

 

where   is cumulative distribution function of the random variable   and   has a uniform distribution over  .   is assumed to be continuous, and   preserves the Lebesgue measure on  .

Polar decomposition of matrices

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When   is a linear map and   is the Gaussian normal distribution, the result coincides with the polar decomposition of matrices. Assuming   where   is an invertible   matrix and considering   the   probability measure, the polar decomposition boils down to

 

where   is a symmetric positive definite matrix, and   an orthogonal matrix. The connection with the polar factorization is   which is convex, and   which preserves the   measure.

Helmholtz decomposition

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The results also allow to recover Helmholtz decomposition. Letting   be a smooth vector field it can then be written in a unique way as

 

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

The connection can be seen by assuming   is the Lebesgue measure on a compact set   and by writing   as a perturbation of the identity map

 

where   is small. The polar decomposition of   is given by  . Then, for any test function   the following holds:

 

where the fact that   was preserving the Lebesgue measure was used in the second equality.

In fact, as  , one can expand  , and therefore  . As a result,   for any smooth function  , which implies that   is divergence-free.[1][6]

See also

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

References

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  1. ^ a b c 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.