On a Neural Implementation of Brenier's Polar Factorization
Nina Vesseron Marco Cuturi
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 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 \citeauthor{Brenier1991PolarFA}'s polar factorization to
non-convex optimization problems, as well as sampling of densities that are not
log-concave.