Transport, Don't Generate: Deterministic Geometric Flows for Combinatorial Optimization

Recent advances in Neural Combinatorial Optimization (NCO) have been dominated by diffusion models that treat the Euclidean Traveling Salesman Problem (TSP) as a stochastic $N \times N$ heatmap generation task. In this paper, we propose CycFlow, a framework that replaces iterative edge denoising with deterministic point transport. CycFlow learns an instance-conditioned vector field that continuously transports input 2D coordinates to a canonical circular arrangement, where the optimal tour is recovered from this $2N$ dimensional representation via angular sorting. By leveraging data-dependent flow matching, we bypass the quadratic bottleneck of edge scoring in favor of linear coordinate dynamics. This paradigm shift accelerates solving speed by up to three orders of magnitude compared to state-of-the-art diffusion baselines, while maintaining competitive optimality gaps.

CanonNet: Canonical Ordering and Curvature Learning for Point Cloud Analysis

Point cloud processing poses two fundamental challenges: establishing consistent point ordering and effectively learning fine-grained geometric features. Current architectures rely on complex operations that limit expressivity while struggling to capture detailed surface geometry. We present CanonNet, a lightweight neural network composed of two complementary components: (1) a preprocessing pipeline that creates a canonical point ordering and orientation, and (2) a geometric learning framework where networks learn from synthetic surfaces with precise curvature values. This modular approach eliminates the need for complex transformation-invariant architectures while effectively capturing local geometric properties. Our experiments demonstrate state-of-the-art performance in curvature estimation and competitive results in geometric descriptor tasks with significantly fewer parameters (\textbf{100X}) than comparable methods. CanonNet's efficiency makes it particularly suitable for real-world applications where computational resources are limited, demonstrating that mathematical preprocessing can effectively complement neural architectures for point cloud analysis. The code for the project is publicly available \hyperlink{https://benjyfri.github.io/CanonNet/}{https://benjyfri.github.io/CanonNet/}.