Sum of Squares Circuits
This work provides a theoretical framework for unifying and separating tractable probabilistic models, which is incremental but important for researchers in probabilistic machine learning.
The paper tackles the expressiveness trade-off in probabilistic circuits by introducing sum of squares PCs, proving they can be exponentially more expressive than existing models like squared and monotonic PCs, and empirically demonstrating their effectiveness in distribution estimation.
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.