On learning higher-order cumulants in diffusion models
This provides theoretical insights into diffusion model training for researchers, though it appears incremental as it builds on existing frameworks.
The paper analyzed how diffusion models learn non-Gaussian correlations by studying higher-order cumulants during forward and backward processes, showing analytically that these correlations are conserved in drift-free models and learned through the score function.
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.