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
Notation. Denote the image measure of through the map .
Definition: Measure preserving map. Let and be some probability spaces and a map. Then, is said to be measure preserving if for every -measurable subset of , is -measurable and , that is: with that is -integrable and that 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
Dimension 1
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
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
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
- polar decomposition – Representation of invertible matrices as unitary operator multiplying a Hermitian operator
References
- ^ 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.
- ^ 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.
- ^ 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.
- ^ Santambrogio, Filippo (2015). Optimal transport for applied mathematicians. New York: Birkäuser. CiteSeerX 10.1.1.726.35.
- ^ 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.
- ^ Villani, Cédric (2003). Topics in optimal transportation. American Mathematical Society.