Applications of optimal transport in multivariate statistics

Finding a desirable generalisation of rank-based statistical methods to multivariate case has been a timeless statistical endeavor. While various concepts of multivariate rank/quantile have been proposed in the past decades, most of them do not maintain crucial properties enjoyed by the traditional univariate rank/quantile, such as distribution-freeness and strong consistency. A novel multivariate rank/quantile from the perspective of optimal transport (OT) was proposed by Chernozhukov et al. [Che+17] and Hallin et al. [Hal+21]. This OT-based concept extends most of desirable properties of traditional rank/quantile on real line to multidimensional space, thus has drawn an increasing attention in recent years. In this essay, we apply the OT-based multivariate rank/quantile on two statistical domains: multiple-output quantile regression and nonparametric independence testing between random vectors. On the first direction, we introduce a robust estimation of multiple-output linear model coefficient by extending the traditional univariate composite quantile regression to the case of multivariate response variable through OT-based techniques. Both the consistency and the convergence rate of the proposed estimator is established under multivariate heavy-tailed random error case. For the second direction, we proposed an geometrically intuitive correlation coefficient for random vectors utilising the OT-based multivariate rank. The proposed coefficient enjoys an entirely distribution-free asymptotic theory under the independent assumption, thus avoiding any permutation-based p-value calculations. Moreover, unlike many existing measurement, the proposed coefficient is capable of detecting not only functional dependency but also spurious correlation via confounders.