# Symmetry-Structured Neural Completion of Islamic Geometric Patterns from Sparse Control Geometry

arXiv: https://arxiv.org/abs/2607.02573 · HTML: https://mlsanity.com/paper/2607.02573/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 37 references that are in the map.
- First posted on arXiv 30 Jun 2026.

## Builds on (most cited first, 16 of 16)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [DiGress: Discrete Denoising diffusion for graph generation](https://mlsanity.com/paper/2209.14734/) (2022)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [GraphRNN: Generating Realistic Graphs with Deep Auto-regressive Models](https://mlsanity.com/paper/1802.08773/) (2018)
- [GraphVAE: Towards Generation of Small Graphs Using Variational Autoencoders](https://mlsanity.com/paper/1802.03480/) (2018)
- [A Neural Representation of Sketch Drawings](https://mlsanity.com/paper/1704.03477/) (2017)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [DeepSVG: A Hierarchical Generative Network for Vector Graphics Animation](https://mlsanity.com/paper/2007.11301/) (2020)
- [Im2Vec: Synthesizing Vector Graphics without Vector Supervision](https://mlsanity.com/paper/2102.02798/) (2021)
- [A Survey of Geometric Graph Neural Networks: Data Structures, Models and Applications](https://mlsanity.com/paper/2403.00485/) (2024)
- [A Neural Tension Operator for Curve Subdivision across Constant Curvature Geometries](https://mlsanity.com/paper/2603.28937/) (2026)

## Abstract (arXiv)

Islamic geometric patterns are governed by exact rotational symmetry and strict construction rules. This paper treats these rules as formal geometric knowledge and embeds them in a neural completion framework, rather than leaving them to be learned statistically from data. Given sparse control geometry and a target symmetry order, the system completes the pattern as a vector graph by predicting edges and refinements of bounded curves over a candidate lattice whose edges are organised into rotational orbits under the cyclic group. Symmetry is enforced either by constraining predictions within these orbits or by projecting them onto them during inference. The orbit-tied variant provides a constructive guarantee: for any input and any orbit-level selection rule, it produces exact N-fold symmetry, preserves anchor points, and keeps all refinements within prescribed bounds. These properties are verified numerically. The study focuses on rotational symmetry, and all quantitative results are obtained from procedurally generated graphs inspired by Islamic geometric design rather than from a historical corpus. On clean inputs, enforcing exact validity produces no measurable loss in fidelity. When control geometry is missing, an unstructured decoder loses fidelity and breaks symmetry; retraining on corrupted inputs recovers much of the fidelity but not exact validity. Symmetry-structured inference, by contrast, keeps violations at zero throughout. The results show that augmentation and symmetry structure address distinct failure modes: augmentation improves fidelity under corruption, while symmetry structure guarantees validity. The framework therefore provides a knowledge-constrained, guarantee-backed approach to neural completion for scalable vector ornaments whose validity depends on exact geometric structure.