# On a Neural Implementation of Brenier's Polar Factorization

AuthorsNina Vesseron, Marco Cuturi Cameto

content type paperpublished July 2024

AuthorsNina Vesseron, Marco Cuturi Cameto

In 1991, Brenier proved a theorem that generalizes the polar 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 explore possible uses within machine learning. The theorem is closely related to optimal transport (OT) theory, and we borrow from recent advances in the field of neural optimal transport to parameterize the potential $u$ as an input convex neural network. The map $M$ can be either evaluated pointwise using $u^*$, the convex conjugate of $u$, through the identity $M=\nabla u^* \circ F$, or learned as an auxiliary network. Because $M$ is, in general, not injective, we consider the additional task of estimating the ill-posed inverse map that can approximate the pre-image measure $M^{-1}$ using a stochastic generator. We illustrate possible applications of Brenier's polar factorization to non-convex optimization problems, as well as sampling of densities that are not log-concave.

Optimal transport (OT) theory focuses, among all maps T:Rd→RdT:\mathbb{R}^d\rightarrow \mathbb{R}^dT:Rd→Rd that can morph a probability measure onto another, on those that are the "thriftiest", i.e. such that the averaged cost c(x,T(x))c(\mathbf{x}, T(\mathbf{x}))c(x,T(x)) between x\mathbf{x}x and its image T(x)T(\mathbf{x})T(x) be as small as possible. Many computational approaches have been proposed to estimate such Monge maps when ccc is the…

See paper detailsOptimal transport (OT) theory has been been used in machine learning to study and characterize maps that can push-forward efficiently a probability measure onto another.
Recent works have drawn inspiration from Brenier's theorem, which states that when the ground cost is the squared-Euclidean distance, the "best" map to morph a continuous measure in P(Rd)\mathcal{P}(\mathbb{R}^d)P(Rd) into another must be the gradient of a convex function.
To…

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