In 1991, Brenier proved a theorem that generalizes the polar decomposition for square matrices — factored as PSD ×times× unitary — to any vector field F:Rd→RdF:mathbb{R}^drightarrow mathbb{R}^dF:Rd→Rd. The theorem, known as the polar factorization theorem, states that any field FFF can be recovered as the composition of the gradient of a convex function uuu with a measure-preserving map MMM, namely F=∇u∘MF=nabla u circ MF=∇u∘M. We propose a practical implementation of this far-reaching theoretical result, and explore possible uses within machine learning. The theorem is closely related…
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