Subdivision Measure Matching: Generative Modeling in Integral Form

Generating samples from distributions with fractal, sharp, sparse, or disconnected support requires reproducing fine-scale structure while preserving distinct modes and gaps. While learned quantizers suffer from codebook collapse or depth saturation, increasing the resolution of scalar grid quantizers leads to exponential growth in the prediction head. To address these limitations, we introduce Subdivision Measure Matching (SMM), which models probability mass through codebook-free recursive subdivision of the domain. SMM progressively refines coarse regions into fine samples, increasing precision exponentially with depth while keeping the categorical prediction dimension fixed. For faster sampling, we introduce parallel coordinate refinement at finer levels, requiring only one model evaluation per level. Theoretically, SMM achieves optimal reconstruction accuracy for its code length and codebook size, with a strict worst-case advantage over shared additive residual quantization, and vanishing factorization-error bounds under standard smoothness assumptions. Experiments across 7 toy distributions demonstrate superior recall, coverage, and depth scaling compared to residual quantization baselines. On D3IL robot imitation tasks, SMM outperforms baselines, including diffusion policies, in 8 of 9 settings and achieves higher behavioral diversity on 3 tasks. On high-dimensional image synthesis, SMM improves ImageNet FID from 15.48 to 13.17 over a Flow Matching baseline.