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.

翻译：推送前向操作使得人们能够通过确定性映射重新分布概率测度。它在统计学和优化中发挥着关键作用：许多学习问题（特别是来自最优传输、生成建模和算法公平性的问题）都包含以模型上的推送前向条件形式呈现的约束或惩罚。然而，文献中缺乏关于此类约束（非）凸性及其对相关学习问题影响的普遍理论见解。本文旨在填补这一空白。在第一部分中，我们提供了两个函数集合（非）凸性的充要条件：将一种概率测度传输到另一种概率测度的映射；以及跨不同概率测度诱导相同输出分布的映射。这揭示了对于大多数概率测度而言，这些推送前向约束并非凸的。在第二部分中，我们展示了这一结果如何对学习生成模型或分组公平预测器的凸优化问题设计构成关键限制。这项工作有望帮助研究人员和实践者更好地理解推送前向条件对凸性的关键影响。