# On the Relationship Between Monotone and Squared Probabilistic Circuits

arXiv: https://arxiv.org/abs/2408.00876 · HTML: https://mlsanity.com/paper/2408.00876/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 60 references that are in the map.
- First posted on arXiv 1 Aug 2024.

## Builds on (most cited first, 30 of 32)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Importance Weighted Autoencoders](https://mlsanity.com/paper/1509.00519/) (2015)
- [Einsum Networks: Fast and Scalable Learning of Tractable Probabilistic Circuits](https://mlsanity.com/paper/2004.06231/) (2020)
- [Semantic Probabilistic Layers for Neuro-Symbolic Learning](https://mlsanity.com/paper/2206.00426/) (2022)
- [A Differential Approach to Inference in Bayesian Networks](https://mlsanity.com/paper/1301.3847/) (2013)
- [Unsupervised Generative Modeling Using Matrix Product States](https://mlsanity.com/paper/1709.01662/) (2017)
- [Tractable Control for Autoregressive Language Generation](https://mlsanity.com/paper/2304.07438/) (2023)
- [Tree Tensor Networks for Generative Modeling](https://mlsanity.com/paper/1901.02217/) (2019)
- [Expressive power of tensor-network factorizations for probabilistic modeling, with applications from hidden Markov models to quantum machine learning](https://mlsanity.com/paper/1907.03741/) (2019)
- [Non-parametric Models for Non-negative Functions](https://mlsanity.com/paper/2007.03926/) (2020)
- [On the Latent Variable Interpretation in Sum-Product Networks](https://mlsanity.com/paper/1601.06180/) (2016)
- [Tractable Regularization of Probabilistic Circuits](https://mlsanity.com/paper/2106.02264/) (2021)
- [Subtractive Mixture Models via Squaring: Representation and Learning](https://mlsanity.com/paper/2310.00724/) (2023)
- [Scaling Up Probabilistic Circuits by Latent Variable Distillation](https://mlsanity.com/paper/2210.04398/) (2022)
- [Tensor-Train Density Estimation](https://mlsanity.com/paper/2108.00089/) (2021)
- [A Unified Approach for Learning the Parameters of Sum-Product Networks](https://mlsanity.com/paper/1601.00318/) (2016)
- [PSD Representations for Effective Probability Models](https://mlsanity.com/paper/2106.16116/) (2021)
- [Scaling Tractable Probabilistic Circuits: A Systems Perspective](https://mlsanity.com/paper/2406.00766/) (2024)
- [Image Inpainting via Tractable Steering of Diffusion Models](https://mlsanity.com/paper/2401.03349/) (2023)
- [Understanding the Distillation Process from Deep Generative Models to Tractable Probabilistic Circuits](https://mlsanity.com/paper/2302.08086/) (2023)
- [On the Expressive Efficiency of Sum Product Networks](https://mlsanity.com/paper/1411.7717/) (2014)
- [Squared Neural Families: A New Class of Tractable Density Models](https://mlsanity.com/paper/2305.13552/) (2023)
- [Building Expressive and Tractable Probabilistic Generative Models: A Review](https://mlsanity.com/paper/2402.00759/) (2024)
- [Polynomial Semantics of Tractable Probabilistic Circuits](https://mlsanity.com/paper/2402.09085/) (2024)
- [Positive semidefinite rank](https://mlsanity.com/paper/1407.4095/) (2014)
- [Compositional Probabilistic and Causal Inference using Tractable Circuit Models](https://mlsanity.com/paper/2304.08278/) (2023)
- [On the Relationship Between Probabilistic Circuits and Determinantal Point Processes](https://mlsanity.com/paper/2006.15233/) (2020)

## Built on by (most cited first, 14 of 14)

- [Sum of Squares Circuits](https://mlsanity.com/paper/2408.11778/) (2024)
- [Scaling Probabilistic Circuits via Monarch Matrices](https://mlsanity.com/paper/2506.12383/) (2025)
- [Restructuring Tractable Probabilistic Circuits](https://mlsanity.com/paper/2411.12256/) (2024)
- [How to Square Tensor Networks and Circuits Without Squaring Them](https://mlsanity.com/paper/2512.17090/) (2025)
- [Tractable Representation Learning with Probabilistic Circuits](https://mlsanity.com/paper/2507.04385/) (2025)
- [Tractable Sharpness-Aware Learning of Probabilistic Circuits](https://mlsanity.com/paper/2508.05537/) (2025)
- [On the Hardness of Approximating Distributions with Tractable Probabilistic Models](https://mlsanity.com/paper/2506.01281/) (2025)
- [The Limits of Tractable Marginalization](https://mlsanity.com/paper/2506.12020/) (2025)
- [Squared families: Searching beyond regular probability models](https://mlsanity.com/paper/2503.21128/) (2025)
- [A Quantum Information Theoretic Approach to Tractable Probabilistic Models](https://mlsanity.com/paper/2506.01824/) (2025)
- [ExplainFuzz: Explainable and Constraint-Conditioned Test Generation with Probabilistic Circuits](https://mlsanity.com/paper/2604.06559/) (2026)
- [How to Approximate Inference with Subtractive Mixture Models](https://mlsanity.com/paper/2604.16714/) (2026)
- [Learning Tractable Distributions Of Language Model Continuations](https://mlsanity.com/paper/2511.16054/) (2025)
- [The Expressivity Boundary of Probabilistic Circuits: A Comparison with Large Language Models](https://mlsanity.com/paper/2605.12940/) (2026)

## Abstract (arXiv)

Probabilistic circuits are a unifying representation of functions as computation graphs of weighted sums and products. Their primary application is in probabilistic modeling, where circuits with non-negative weights (monotone circuits) can be used to represent and learn density/mass functions, with tractable marginal inference. Recently, it was proposed to instead represent densities as the square of the circuit function (squared circuits); this allows the use of negative weights while retaining tractability, and can be exponentially more expressive efficient than monotone circuits. Unfortunately, we show the reverse also holds, meaning that monotone circuits and squared circuits are incomparable in general. This raises the question of whether we can reconcile, and indeed improve upon the two modeling approaches. We answer in the positive by proposing Inception PCs, a novel type of circuit that naturally encompasses both monotone circuits and squared circuits as special cases, and employs complex parameters. Empirically, we validate that Inception PCs can outperform both monotone and squared circuits on a range of tabular and image datasets.