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, naïvely 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.
Figures & tables
Figure 1: Mean Test Log-likelihood ( ↑ ) on synthetic 2D and 3D density estimation tasks achieved by EinsumNet and HCLT along with their geometry-aware extensions using Voronoi tessellations (VT) and hierarchical factorized Voronoi (HFV), averaged across 3 trials. For VT, values correspond to the lower bound on the log-likelihood obtained via our certified approximate inference framework.
Figure 2: Visualization of the distribution and voronoi tessellations learned by a VT-EinsumNet (left) and HFV-EinsumNet (right) on the 2D pinwheel dataset. The axis aligned boxes in the left figure represent the Inner Boxes computed for estimating the lower bound on the partition function using our conservative construction.
Figure 3: Learning curves on the 2D spiral dataset. Validation log-likelihood across epochs for EinsumNet (left) and HCLT (right), averaged over 3 trials. VT models report a certified lower bound while baselines and HFV use exact inference.
Figure 4: Learning curves on the UCI tabular datasets. Validation log-likelihood across epochs on UCI tabular benchmarks (EinsumNet family), ordered by increasing dimensionality. VT models report a certified lower bound while baselines and HFV use exact inference.
Model
power
gas
hepmass
miniboone
(6D)
(8D)
(21D)
(43D)
EinsumNet
0.56
5.85
−22.35
−32.24
HFV-EinsumNet
0.56
5.86
−22.32
−32.18
VT-EinsumNet [lo]
1.73
7.13
−22.98
−32.82
VT-EinsumNet [up]
3.46
9.76
−4.82
−14.07
HCLT
0.55
7.22
−20.36
−25.43
Table 1: Mean test log-likelihood ( ↑ ) on UCI tabular benchmarks, averaged over 3 trials. VT rows report certified lower [lo] and upper [up] bounds without additional refinement.
Appendix figures & tables1 asset
Supplementary material from the paper’s appendix.
Appendix
Figure 5: Synthetic 2D/3D density estimation benchmarks. Top row: 2D datasets with diverse local geometry and disconnected support. Bottom row: 3D manifold-like datasets exhibiting crossings, interlocks, and knotting. These benchmarks stress-test whether geometry-aware routing can specialize locally while maintaining reliable inference (VT via certification; HFV via alignment).
Probabilistic circuits are structured generative models that enable tractable probabilistic inference while balancing expressivity and computational cost.
We introduce VoroFields, a hierarchical neural-field framework for approximating generalized Voronoi diagrams of finite geometric site sets in low-dimensional domains under arbitrary evaluable point-to-site distances. Instead of constructing the diagram combinatorially, VoroFields learns a continuous, differentiable surrogate whose maximizer structure induces the partition implicitly. The Voronoi cells correspond to maximizer regions of the field, with boundaries defined by equal responses between competing sites. A hierarchical decomposition reduces the combinatorial complexity by refining only near envelope transition strata. Experiments across site families and metrics demonstrate accurate recovery of cells and boundary geometry without shape-specific constructions.
Panagiotis Rigas, George Ioannakis, Ioannis Emiris
Department of Informatics & Telecommunications, National and Kapodistrian University of Athens, Greece · Archimedes, Athena Research Center, Greece · Institute for Language and Speech Processing, Athena Research Center, Greece
Probabilistic circuits (PCs) can model complex joint distributions while supporting exact and efficient computation of many inference queries. However, standard likelihood-based PC learning is vulnerable to overfitting and fragile generalization when confronted with data noise, small sample sizes, or distribution shifts. This can be mitigated using distributionally-robust optimization which consider worst-case distributions within a Wasserstein ball of the empirical distribution, but current methods are limited to training a model from scratch in this framework. Instead, we propose PeTeR: a novel, data-free post-training framework designed to robustify pre-trained PCs against distribution shifts without retraining from scratch. Empirical evaluations across multiple density estimation benchmarks demonstrate that PeTeR effectively robustifies baseline models against both random and adversarial perturbations, achieving competitive or superior performance to data-dependent robust learning baselines.
Adrian Ciotinga, Yeming Dai, YooJung Choi
School of Computing and Augmented Intelligence, Arizona State University
Vanilla variational inference finds an optimal approximation to the Bayesian posterior distribution, but even the exact Bayesian posterior is often not meaningful under model misspecification. We propose predictive variational inference (PVI): a general inference framework that seeks and samples from an optimal posterior density such that the resulting posterior predictive distribution is as close to the true data generating process as possible, while this closeness is measured by multiple scoring rules. By optimizing the objective, the predictive variational inference is generally not the same as, or even attempting to approximate, the Bayesian posterior, even asymptotically. Rather, we interpret it as implicit hierarchical expansion. Further, the learned posterior uncertainty detects heterogeneity of parameters among the population, enabling automatic model diagnosis. This framework applies to both likelihood-exact and likelihood-free models. We demonstrate its application in real data examples.
Jinlin Lai, Antonio Linero, Yuling Yao
College of Information and Computer Sciences, University of Massachusetts Amherst · Department of Statistics and Data Sciences, University of Texas, Austin