Risk-Sensitive Diffusion: Learning the Underlying Distribution from Noisy Samples

While achieving remarkable performances, we show that diffusion models are fragile to the presence of noisy samples, limiting their potential in the vast amount of settings where, unlike image synthesis, we are not blessed with clean data. Motivated by our finding that such fragility originates from the distribution gaps between noisy and clean samples along the diffusion process, we introduce risk-sensitive SDE, a stochastic differential equation that is parameterized by the risk (i.e., data "dirtiness") to adjust the distributions of noisy samples, reducing misguidance while benefiting from their contained information. The optimal expression for risk-sensitive SDE depends on the specific noise distribution, and we derive its parameterizations that minimize the misguidance of noisy samples for both Gaussian and general non-Gaussian perturbations. We conduct extensive experiments on both synthetic and real-world datasets (e.g., medical time series), showing that our model effectively recovers the clean data distribution from noisy samples, significantly outperforming conditional generation baselines.