Can standard continuous-time generative models represent distributions whose support is an extremely sparse, globally constrained discrete set? We study this question using completed Sudoku grids as a controlled testbed, treating them as a subset of a continuous relaxation space. We train flow-matching and score-based models along a Gaussian probability path and compare deterministic (ODE) sampling, stochastic (SDE) sampling, and DDPM-style discretizations derived from the same continuous-time training. Unconditionally, stochastic sampling substantially outperforms deterministic flows; score-based samplers are the most reliable among continuous-time methods, and DDPM-style ancestral sampling achieves the highest validity overall. We further show that the same models can be repurposed for guided generation: by repeatedly sampling completions under clamped clues and stopping when constraints are satisfied, the model acts as a probabilistic Sudoku solver. Although far less sample-efficient than classical solvers and discrete-geometry-aware diffusion methods, these experiments demonstrate that classic diffusion/flow formulations can assign non-zero probability mass to globally constrained combinatorial structures and can be used for constraint satisfaction via stochastic search.

翻译：标准连续时间生成模型能否表示支撑集为极度稀疏、全局约束的离散集合的分布？我们以已完成的数独网格作为受控测试平台来研究此问题，将其视为连续松弛空间的子集。我们沿高斯概率路径训练流匹配模型和基于分数的模型，并比较源自同一连续时间训练的确定性（ODE）采样、随机性（SDE）采样以及DDPM风格离散化方法。在无条件生成中，随机采样显著优于确定性流；基于分数的采样器在连续时间方法中最为可靠，而DDPM风格的祖先采样总体上获得了最高的有效性。我们进一步表明，相同模型可被重新用于引导生成：通过在固定线索下重复采样补全结果并在约束满足时停止，该模型可充当概率性数独求解器。尽管这些方法的样本效率远低于经典求解器及具有离散几何感知的扩散方法，但实验证明经典扩散/流公式能够为非零概率质量分配至全局约束的组合结构，并可通过随机搜索用于约束满足问题。