Consider a heterogeneous population of points evolving with time. While the population evolves, both in size and nature, we can observe it periodically, through snapshots taken at different timestamps. Each of these snapshots is formed by sampling points from the population at that time, and then creating features to recover point clouds. While these snapshots describe the population's evolution on aggregate, they do not provide directly insights on individual trajectories. This scenario is encountered in several applications, notably single-cell genomics experiments, tracking of particles, or when studying crowd motion. In this paper, we propose to model that dynamic as resulting from the celebrated Jordan-Kinderlehrer-Otto (JKO) proximal scheme. The JKO scheme posits that the configuration taken by a population at time $t$ is one that trades off a decrease w.r.t. an energy (the model we seek to learn) penalized by an optimal transport distance w.r.t. the previous configuration. To that end, we propose JKOnet, a neural architecture that combines an energy model on measures, with (small) optimal displacements solved with input convex neural networks (ICNN). We demonstrate the applicability of our model to explain and predict population dynamics.
翻译:尽管人口在大小和性质上都有变化,但我们可以通过在不同时间戳上拍摄的快照来定期观察。这些快照都是由当时人口抽样点形成,然后创造恢复点云的特征。虽然这些快照描述了人口的总体演变,但没有直接洞察到单个轨迹。在几个应用中,特别是单细胞基因组实验、粒子跟踪或研究人群运动时,都遇到这种假设。在本文中,我们提议以约旦-辛德勒-元首-奥托(JKO)所庆祝的原始计划为模型来模拟这种动态。JKO方案假设,一个人口在时间值上采用的配置是一种交换下降的能源(我们试图学习的模型),它受到最佳运输距离w.r.t.t.的制约。为此,我们提议建立一个神经结构,将能源模型与测量的模型结合起来,同时(小)优化的人口迁移模型与我们输入的恒定动力网络(我们用输入的恒定的动力网络)解析。