Conservation Laws for Diffusion Models

While autoregressive models optimize the exact likelihood implied by the chain rule, diffusion models are typically trained with denoising objectives. We develop conservation laws based on generalized extrinsic information transfer (GEXIT) functions for a broad class of memoryless noise processes, showing that the data--model cross-entropy (CE) can be characterized \emph{exactly} as an integral of \emph{local} information-theoretic derivatives along the noise path. This yields a unified characterization of the likelihood for discrete diffusion and continuous diffusion, with the Gaussian case reducing to the well-known I-MMSE relationship connecting mutual information and estimation theory. An immediate implication is a \emph{locality} property: one can compute the information-theoretic derivatives using only the marginal posteriors along the noise path. As a result, training reduces to learning the marginal posteriors by minimizing the negative log-likelihood. While the conservation law implies that the entropy does not depend on the noise path, finite-capacity denoisers approximate the posteriors with varying accuracy across noise types, leading to differences in performance. We validate these predictions on synthetic Markov sources and standard benchmarks, including \texttt{text8} and CIFAR-10.