Designing expressive generative models that support exact and efficient inference is a core question in probabilistic ML. Probabilistic circuits (PCs) offer a framework where this tractability-vs-expressiveness trade-off can be analyzed theoretically. Recently, squared PCs encoding subtractive mixtures via negative parameters have emerged as tractable models that can be exponentially more expressive than monotonic PCs, i.e., PCs with positive parameters only. In this paper, we provide a more precise theoretical characterization of the expressiveness relationships among these models. First, we prove that squared PCs can be less expressive than monotonic ones. Second, we formalize a novel class of PCs -- sum of squares PCs -- that can be exponentially more expressive than both squared and monotonic PCs. Around sum of squares PCs, we build an expressiveness hierarchy that allows us to precisely unify and separate different tractable model classes such as Born Machines and PSD models, and other recently introduced tractable probabilistic models by using complex parameters. Finally, we empirically show the effectiveness of sum of squares circuits in performing distribution estimation.

本研究解决了概率机器学习中设计高效推理的生成模型的挑战，提出了一种名为平方和电路的新型概率电路。这种电路在表达能力上能够显著超越传统的单调电路，并开发了一种表达能力层次结构，能够清晰地区分和统一不同的可处理模型类，最终通过实证分析展示了平方和电路在分布估计中的有效性。