A methodology is developed to extract $d$ invariant features $W=f(X)$ that predict a response variable $Y$ without being confounded by variables $Z$ that may influence both $X$ and $Y$. The methodology's main ingredient is the penalization of any statistical dependence between $W$ and $Z$ conditioned on $Y$, replaced by the more readily implementable plain independence between $W$ and the random variable $Z_Y = T(Z,Y)$ that solves the [Monge] Optimal Transport Barycenter Problem for $Z\mid Y$. In the Gaussian case considered in this article, the two statements are equivalent. When the true confounders $Z$ are unknown, other measurable contextual variables $S$ can be used as surrogates, a replacement that involves no relaxation in the Gaussian case if the covariance matrix $Σ_{ZS}$ has full range. The resulting linear feature extractor adopts a closed form in terms of the first $d$ eigenvectors of a known matrix. The procedure extends with little change to more general, non-Gaussian / non-linear cases.

Given a basis space (e.g., pixel grid) and a transportation cost function (e.g.,

Optimal transport (OT) barycenters are a mathematically grounded way of averaging probability distributions while capturing their geometric properties.

We present a novel algorithm to estimate the barycenter of arbitrary probability distributions with respect to the Sinkhorn divergence. Based on a Frank-Wolfe optimization strategy, our approach proceeds by populating the support of the barycenter incrementally, without requiring any pre-allocation.

Neural Information Processing Systems http://nips.cc/

The goal of optimal transport is to find a minimal cost of moving masses between (supports of) probability distributions. It is known that the estimation of transport cost is not robust when there are outliers.

In statistics and machine learning, it is often desirable to aggregate distinct but similar collections of information, represented as probability distributions.

Optimal transport (OT) barycenters are a mathematically grounded way of averaging probability distributions while capturing their geometric properties.