# Characteristic Circuits

arXiv: https://arxiv.org/abs/2312.07790 · HTML: https://mlsanity.com/paper/2312.07790/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 49 references that are in the map.
- First posted on arXiv 12 Dec 2023.

## Builds on (most cited first, 18 of 18)

- [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)
- [Fast Two-Sample Testing with Analytic Representations of Probability Measures](https://mlsanity.com/paper/1506.04725/) (2015)
- [The Randomized Dependence Coefficient](https://mlsanity.com/paper/1304.7717/) (2013)
- [A Characteristic Function Approach to Deep Implicit Generative Modeling](https://mlsanity.com/paper/1909.07425/) (2019)
- [Probabilistic Generating Circuits](https://mlsanity.com/paper/2102.09768/) (2021)
- [Bayesian Learning of Sum-Product Networks](https://mlsanity.com/paper/1905.10884/) (2019)
- [Strudel: Learning Structured-Decomposable Probabilistic Circuits](https://mlsanity.com/paper/2007.09331/) (2020)
- [Interventional Sum-Product Networks: Causal Inference with Tractable Probabilistic Models](https://mlsanity.com/paper/2102.10440/) (2021)
- [Probabilistic Circuits for Variational Inference in Discrete Graphical Models](https://mlsanity.com/paper/2010.11446/) (2020)
- [Continuous Mixtures of Tractable Probabilistic Models](https://mlsanity.com/paper/2209.10584/) (2022)
- [Deep Structured Mixtures of Gaussian Processes](https://mlsanity.com/paper/1910.04536/) (2019)
- [Tractable Uncertainty for Structure Learning](https://mlsanity.com/paper/2204.14170/) (2022)
- [Automatic Bayesian Density Analysis](https://mlsanity.com/paper/1807.09306/) (2018)
- [The Sum-Product Theorem: A Foundation for Learning Tractable Models](https://mlsanity.com/paper/1611.03553/) (2016)
- [Leveraging Probabilistic Circuits for Nonparametric Multi-Output Regression](https://mlsanity.com/paper/2106.08687/) (2021)
- [Certifying Fairness of Probabilistic Circuits](https://mlsanity.com/paper/2212.02474/) (2022)

## Built on by (most cited first, 8 of 8)

- [What is the Relationship between Tensor Factorizations and Circuits (and How Can We Exploit it)?](https://mlsanity.com/paper/2409.07953/) (2024)
- [On the Relationship Between Monotone and Squared Probabilistic Circuits](https://mlsanity.com/paper/2408.00876/) (2024)
- [Building Expressive and Tractable Probabilistic Generative Models: A Review](https://mlsanity.com/paper/2402.00759/) (2024)
- [How to Square Tensor Networks and Circuits Without Squaring Them](https://mlsanity.com/paper/2512.17090/) (2025)
- [chiSPN: Characteristic Interventional Sum-Product Networks for Causal Inference in Hybrid Domains](https://mlsanity.com/paper/2408.07545/) (2024)
- [The Limits of Tractable Marginalization](https://mlsanity.com/paper/2506.12020/) (2025)
- [Approximate Bayesian Inference via Bitstring Representations](https://mlsanity.com/paper/2508.13598/) (2025)
- [Variance Computation for Weighted Model Counting with Knowledge Compilation Approach](https://mlsanity.com/paper/2601.03523/) (2026)

## Abstract (arXiv)

In many real-world scenarios, it is crucial to be able to reliably and efficiently reason under uncertainty while capturing complex relationships in data. Probabilistic circuits (PCs), a prominent family of tractable probabilistic models, offer a remedy to this challenge by composing simple, tractable distributions into a high-dimensional probability distribution. However, learning PCs on heterogeneous data is challenging and densities of some parametric distributions are not available in closed form, limiting their potential use. We introduce characteristic circuits (CCs), a family of tractable probabilistic models providing a unified formalization of distributions over heterogeneous data in the spectral domain. The one-to-one relationship between characteristic functions and probability measures enables us to learn high-dimensional distributions on heterogeneous data domains and facilitates efficient probabilistic inference even when no closed-form density function is available. We show that the structure and parameters of CCs can be learned efficiently from the data and find that CCs outperform state-of-the-art density estimators for heterogeneous data domains on common benchmark data sets.