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关联性，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。我们推导了矩生成泛函与累积量生成泛函的显式表达式，其形式取决于初始数据分布及正向过程的特性。解析结果表明：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，因此正向过程的终态仍保留非平凡关联性。我们证明由于这些关联性被编码在得分函数中，即使从正态先验出发，高阶累积量也能在反向过程中被有效学习。我们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验证了上述解析结论。