Control Laguerre Tessellation: Semi-discrete Optimal Transport Over Control Systems

What the math gives to ML

The paper generalizes semi-discrete optimal transport from ordinary geometric costs to costs generated by optimal-control problems, while preserving a Laguerre-cell characterization under a twist condition. The transferable asset is a structured hard assignment rule: each target prototype owns a control-cost Laguerre region, and dual weights adjust those regions to meet prescribed capacities. This suggests replacing Euclidean nearest-centroid routing in mixture-of-experts, vector quantization, or prototype models with dynamics-aware routing whose regions encode the cost of moving an embedding through a controllable latent system. The most practical first test is a balanced MoE router using a linear minimum-energy control cost and online dual updates for exact expert utilization.