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Apple Machine Learning Research |
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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}^d\rightarrow \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…