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Causal models are fundamental tools to understand complex systems and predict the effect of interventions on such systems. However, despite an extensive literature in the population (or infinite-sample) case, where distributions are assumed to be known, little is known about the statistical rates of convergence of various methods, even for the simplest models. In this work, allowing for cycles, we study linear structural equations models with homoscedastic Gaussian noise and in the presence of interventions that make the model identifiable. More specifically, we present statistical rates of estimation for both the LLC estimator introduced by Hyttinen, Eberhardt and Hoyer and a novel two-step penalized maximum likelihood estimator. We establish asymptotic near minimax optimality for the maximum likelihood estimator over a class of sparse causal graphs in the case of near-optimally chosen interventions. Moreover, we find evidence for practical advantages of this estimator compared to LLC in synthetic numerical experiments. 

We propose a new method to estimate Wasserstein distances and optimal transport plans between two probability distributions from samples in high dimension. Unlike plug-in rules that simply replace the true distributions by their empirical counterparts, our method promotes couplings with low transport rank, a new structural assumption that is similar to the nonnegative rank of a matrix. Regularizing based on this assumption leads to drastic improvements on high-dimensional data for various tasks, including domain adaptation in single-cell RNA sequencing data. These findings are supported by a theoretical analysis that indicates that the transport rank is key in overcoming the curse of dimensionality inherent to data-driven optimal transport.