Machine Learning in Montpellier, Theory & Practice – Luca Ganassali (Univ. Paris-Saclay)
Coupling probability measures is central to statistics and machine learning, yet transport maps are generally non-unique. The common recourse to optimal transport, motivated by cost minimization and cyclical monotonicity, obscures the fact that several distinct notions of multivariate monotone matchings coexist. In this talk, we will first compare three constructions of transport maps—cyclically monotone, quantile-preserving, and triangular maps—characterizing when they coincide and highlighting their structural properties. We will then connect this analysis to causal inference, showing how counterfactual reasoning can be framed as selecting a transport map and when causal assumptions align with classical statistical transports. Taken together, these results aim to enrich the theoretical understanding of families of transport maps and to clarify their possible causal interpretations. This talk is based on joint work with Lucas De Lara.