# Geometry-Aware Probabilistic Circuits via Voronoi Tessellations

arXiv: https://arxiv.org/abs/2603.11946 · HTML: https://mlsanity.com/paper/2603.11946/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 45 references that are in the map.
- First posted on arXiv 12 Mar 2026.

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

- [Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer](https://mlsanity.com/paper/1701.06538/) (2017)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Einsum Networks: Fast and Scalable Learning of Tractable Probabilistic Circuits](https://mlsanity.com/paper/2004.06231/) (2020)
- [A Differential Approach to Inference in Bayesian Networks](https://mlsanity.com/paper/1301.3847/) (2013)
- [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)
- [Gated Linear Networks](https://mlsanity.com/paper/1910.01526/) (2019)
- [A Unified Approach for Learning the Parameters of Sum-Product Networks](https://mlsanity.com/paper/1601.00318/) (2016)
- [Scaling Tractable Probabilistic Circuits: A Systems Perspective](https://mlsanity.com/paper/2406.00766/) (2024)
- [Interventional Sum-Product Networks: Causal Inference with Tractable Probabilistic Models](https://mlsanity.com/paper/2102.10440/) (2021)
- [What is the Relationship between Tensor Factorizations and Circuits (and How Can We Exploit it)?](https://mlsanity.com/paper/2409.07953/) (2024)
- [Sum of Squares Circuits](https://mlsanity.com/paper/2408.11778/) (2024)
- [Continuous Mixtures of Tractable Probabilistic Models](https://mlsanity.com/paper/2209.10584/) (2022)
- [Understanding the Distillation Process from Deep Generative Models to Tractable Probabilistic Circuits](https://mlsanity.com/paper/2302.08086/) (2023)
- [Sparse Probabilistic Circuits via Pruning and Growing](https://mlsanity.com/paper/2211.12551/) (2022)
- [HyperSPNs: Compact and Expressive Probabilistic Circuits](https://mlsanity.com/paper/2112.00914/) (2021)
- [Building Expressive and Tractable Probabilistic Generative Models: A Review](https://mlsanity.com/paper/2402.00759/) (2024)
- [Semi-Discrete Normalizing Flows through Differentiable Tessellation](https://mlsanity.com/paper/2203.06832/) (2022)
- [VQ-Flows: Vector Quantized Local Normalizing Flows](https://mlsanity.com/paper/2203.11556/) (2022)
- [Scaling Probabilistic Circuits via Monarch Matrices](https://mlsanity.com/paper/2506.12383/) (2025)
- [Probabilistic Integral Circuits](https://mlsanity.com/paper/2310.16986/) (2023)
- [Voronoi Density Estimator for High-Dimensional Data: Computation, Compactification and Convergence](https://mlsanity.com/paper/2206.08051/) (2022)
- [Probabilistic Circuits That Know What They Don't Know](https://mlsanity.com/paper/2302.06544/) (2023)
- [Probabilistic Neural Circuits](https://mlsanity.com/paper/2403.06235/) (2024)
- [Sum-Product-Quotient Networks](https://mlsanity.com/paper/1710.04404/) (2017)
- [Credibility-Aware Multi-Modal Fusion Using Probabilistic Circuits](https://mlsanity.com/paper/2403.03281/) (2024)
- [A Unified Framework for Human-Allied Learning of Probabilistic Circuits](https://mlsanity.com/paper/2405.02413/) (2024)
- [Rethinking Probabilistic Circuit Parameter Learning](https://mlsanity.com/paper/2505.19982/) (2025)
- [Tractable Representation Learning with Probabilistic Circuits](https://mlsanity.com/paper/2507.04385/) (2025)

## Built on by (most cited first, 1 of 1)

- [A Compositional Theory of Curvature in Probabilistic Circuits](https://mlsanity.com/paper/2608.12869/) (2026)

## Abstract (arXiv)

Probabilistic circuits (PCs) enable exact and tractable inference but employ data independent mixture weights that limit their ability to capture local geometry of the data manifold. We propose Voronoi tessellations (VT) as a natural way to incorporate geometric structure directly into the sum nodes of a PC. However, na"ively introducing such structure breaks tractability. We formalize this incompatibility and develop two complementary solutions: (1) an approximate inference framework that provides guaranteed lower and upper bounds for inference, and (2) a structural condition for VT under which exact tractable inference is recovered. Finally, we introduce a differentiable relaxation for VT that enables gradient-based learning and empirically validate the resulting approach on standard density estimation tasks.