Wasserstein gradient flow for optimal probability measure decomposition

With the rapid advancement of AI, automated algorithms are increasingly being used to solve routine problems. Particularly intriguing are the applications of AI in social organizations, which have the potential to benefit both private and public sectors. These applications include the organization of markets, allocation of resources, and mechanism design, among others (Agrawal et al. 2023, Chen et al. 2021, Dai and Jordan 2021, Niazadeh et al. 2023, Zhalechian et al. 2022). This paper studies a new problem of how to decompose a population of customers or clients into groups to optimize a generic quantitive criterion. Consider the following probability measure decomposition problem. Later, we will show how this problem can arise in applications.