The push-forward operation enables one to redistribute a probability measure
through a deterministic map. It plays a key role in statistics and
optimization: many learning problems (notably from optimal transport,
generative modeling, and algorithmic fairness) include constraints or penalties
framed as push-forward conditions on the model. However, the literature lacks
general theoretical insights on the (non)convexity of such constraints and its
consequences on the associated learning problems. This paper aims at filling
this gap. In a first part, we provide a range of sufficient and necessary
conditions for the (non)convexity of two sets of functions: the maps
transporting one probability measure to another; the maps inducing equal output
distributions across distinct probability measures. This highlights that for
most probability measures, these push-forward constraints are not convex. In a
second time, we show how this result implies critical limitations on the design
of convex optimization problems for learning generative models or group-fair
predictors. This work will hopefully help researchers and practitioners have a
better understanding of the critical impact of push-forward conditions onto
convexity.

通过推限操作，可以通过确定性映射重新分配概率度量，从而在统计学和优化中发挥着关键作用。本文旨在填补相关领域（最优传输、生成模型和算法公平性等）中关于这些约束的凸性（非凸性）及其对相关学习问题的影响的缺乏理论洞察力的现状。