To analyse how diffusion models learn correlations beyond Gaussian ones, we study the behaviour of higher-order cumulants, or connected n-point functions, under both the forward and backward process. We derive explicit expressions for the moment- and cumulant-generating functionals, in terms of the distribution of the initial data and properties of forward process. It is shown analytically that during the forward process higher-order cumulants are conserved in models without a drift, such as the variance-expanding scheme, and that therefore the endpoint of the forward process maintains nontrivial correlations. We demonstrate that since these correlations are encoded in the score function, higher-order cumulants are learnt in the backward process, also when starting from a normal prior. We confirm our analytical results in an exactly solvable toy model with nonzero cumulants and in scalar lattice field theory.

翻译：为分析扩散模型如何学习超越高斯分布的相关性，我们研究了高阶累积量（即连通n点函数）在正向与反向过程中的演化行为。通过初始数据分布与正向过程特性的函数关系，我们推导了矩生成泛函与累积量生成泛函的显式表达式。理论分析表明：在无漂移项的模型（如方差扩展方案）中，高阶累积量在正向过程中保持守恒，因此正向过程的终点仍保持非平凡相关性。我们证明这些相关性被编码在得分函数中，使得即使从正态先验出发，反向过程仍能学习高阶累积量。通过具有非零累积量的精确可解玩具模型及标量格点场论，我们验证了理论分析结果。