Probabilistic integral circuits (PICs) have been recently introduced as probabilistic models enjoying the key ingredient behind expressive generative models: continuous latent variables (LVs). PICs are symbolic computational graphs defining continuous LV models as hierarchies of functions that are summed and multiplied together, or integrated over some LVs. They are tractable if LVs can be analytically integrated out, otherwise they can be approximated by tractable probabilistic circuits (PC) encoding a hierarchical numerical quadrature process, called QPCs. So far, only tree-shaped PICs have been explored, and training them via numerical quadrature requires memory-intensive processing at scale. In this paper, we address these issues, and present: (i) a pipeline for building DAG-shaped PICs out of arbitrary variable decompositions, (ii) a procedure for training PICs using tensorized circuit architectures, and (iii) neural functional sharing techniques to allow scalable training. In extensive experiments, we showcase the effectiveness of functional sharing and the superiority of QPCs over traditional PCs.

翻译：概率积分电路（PICs）近期被提出作为概率模型，其具备生成式表达模型的关键要素：连续潜变量（LVs）。PICs是符号计算图，通过函数的求和、乘积或对某些LVs的积分来定义连续LV模型的分层结构。若LVs可解析积分，则PICs是易于处理的；否则，可通过称为QPCs的可处理概率电路（PC）进行近似，该电路编码分层数值求积过程。迄今仅探索了树形PICs，而通过数值求积训练此类模型在扩展时需高内存处理。本文针对这些问题提出：（i）基于任意变量分解构建有向无环图（DAG）形PICs的流程，（ii）利用张量化电路架构训练PICs的方法，以及（iii）实现可扩展训练的函数共享技术。通过大量实验，我们展示了函数共享的有效性以及QPCs相较传统PCs的优越性。