cs.LGMar 12, 2026

Geometry-Aware Probabilistic Circuits via Voronoi Tessellations

Authors: Sahil Sidheekh, Sriraam Natarajan

Organizations: The University of Texas at Dallas, Richardson, TX, USA.

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Abstract

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 ( ↑\uparrow ) 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 33 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 2D2D 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 33 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.
Modelpowergashepmassminiboone
 (6D)(8D)(21D)(43D)
EinsumNet0.560.565.855.85−22.35-22.35−32.24-32.24
HFV-EinsumNet0.560.565.865.86−22.32\bm{-22.32}−32.18\bm{-32.18}
VT-EinsumNet [lo]1.73\bm{1.73}7.13\bm{7.13}−22.98-22.98−32.82-32.82
VT-EinsumNet [up]3.463.469.769.76−4.82-4.82−14.07-14.07
HCLT0.550.557.227.22−20.36-20.36−25.43-25.43
Table 1: Mean test log-likelihood ( ↑\uparrow ) 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).
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