On a Neural Implementation of Brenier's Polar Factorization

arXiv:2403.03071v1 Announce Type: cross
Abstract: In 1991, Brenier proved a theorem that generalizes the $QR$ decomposition for square matrices -- factored as PSD $\times$ unitary -- to any vector field $F:\mathbb{R}^d\rightarrow \mathbb{R}^d$. The theorem, known as the polar factorization theorem, states that any field $F$ can be recovered as the composition of the gradient of a convex function $u$ with a measure-preserving map $M$, namely $F=\nabla u \circ M$. We propose a practical implementation of this far-reaching theoretical result, and …

abstract arxiv cs.lg factorization implementation polar square stat.ml theorem type vector