Brenier's Polar Factorization Theore
Brenier's polar factorization theorem, developed by Brenier (1987) in the field of optimal transport, is a fundamental result in mathematics. It involves the polar decomposition of vector fields and is related to other theories such as Monge-Kantorovich theory. In physical terms, the theorem can be understood as stating that quadratically optimal transport plans move a physical quantity with minimal amount of energy. The theorem has been proven using different methods, including Euler-Lagrange equations and the Monge-Ampère equation, and has been extended to various scenarios, such as nondegeneracy conditions and the transformation of trained neural networks. The theory is also connected to the study of Borel maps and bounded domains.
Brenier's polar factorization theorem, formulated by Brenier in 1987, is a significant result in the realm of optimal transport. It involves the polar decomposition of vector fields and has implications for various theories such as Monge-Kantorovich theory. In practical terms, the theorem signifies that quadratically optimal transport plans allow the movement of a physical quantity with minimal energy expenditure. The theorem has been substantiated using diverse methods, including the utilization of Euler-Lagrange equations and the Monge-Ampère equation. It has also been extended to encompass a range of scenarios, such as nondegeneracy conditions and the transformation of trained neural networks. This theory is intricately linked to the study of Borel maps and bounded domains.
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