Martingale Residuals for Evaluating and Training Diffusion Models

Diffusion models are typically trained with noise or score targets. We introduce a martingale residual (MR) framework for evaluation and learning from sampled reference paths, without requiring these targets. A martingale residual is the observed change along a reference path minus the change predicted by the model. We prove that, under suitable assumptions, MR orthogonality implies that the learned and reference processes have the same path law. We combine weighted residual expectations into an MR score. When the learned process uses the reference process's known covariance, the MR score formed from all coordinate test functions converges to the squared drift error as the time interval shrinks, provided the instruments form a complete orthonormal basis. Finite instruments give a projection approximation. For DDPMs, suitable residual scaling connects this limit to denoising MSE minus a constant independent of the model parameters. We evaluate the MR score as a diagnostic, a training regularizer, and a sole training objective. On CIFAR-10, MR scores track denoising training, with richer instrument sets capturing a larger fraction of the denoising MSE decrease. MR regularization yields modest improvements in sample quality during continuation of pretrained CIFAR-10 and ImageNet-64 models. Using the MR score alone, we learn stochastic transport from a Gaussian mixture to a two-moons distribution in a Schrödinger bridge task.