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Geometry-Aware Probabilistic Circuits via Voronoi Tessellations

Sahil Sidheekh, Sriraam Natarajan

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.

Geometry-Aware Probabilistic Circuits via Voronoi Tessellations

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.
Paper Structure (36 sections, 23 theorems, 55 equations, 4 figures, 3 algorithms)

This paper contains 36 sections, 23 theorems, 55 equations, 4 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background & Related Work
  3. Geometry-Aware Probabilistic Circuits
  4. Certified Approximate Inference
  5. Anytime Bound Refinement
  6. Propagating Bounds Through the Circuit.
  7. Hierarchical Factorized Voronoi PCs
  8. Learning via Soft Gating
  9. Experiments & Results
  10. Conclusion
  11. Proofs of Main Results
  12. Intractability of Voronoi-Gated PCs
  13. Deep Circuits and Constraint Accumulation
  14. Geometric Alignment and Tractability Recovery
  15. Certified Approximate Inference
  16. ...and 21 more sections

Key Result

Proposition 3.2

Let $f(\mathbf{x})=\sum_{k=1}^K g_k(\mathbf{x})\,\pi_k\,p_k(\mathbf{x})$ be a Voronoi-gated sum node over $\mathbf{X}=\{X_1,\ldots,X_D\}$. Suppose each child is fully factorized, $p_k(\mathbf{x})=\prod_{i=1}^D p_k^{(i)}(x_i)$. Then the partition function $Z=\int f(\mathbf{x})\,d\mathbf{x}=\sum_{k=1}

Figures (4)

  • 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 $3$ 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 log-likelihood lower bound while baselines and HFV use exact tractable evaluation.
  • Figure 4: 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).

Theorems & Definitions (51)

  • Definition 2.1
  • Definition 2.2: Scope
  • Definition 2.3: Smoothness
  • Definition 2.4: Decomposability
  • Definition 2.5: Determinism
  • Definition 3.1
  • Proposition 3.2: Single-Layer Intractability
  • Remark 3.3
  • Definition 3.4: Geometric Alignment
  • Theorem 3.5
  • ...and 41 more