When can diffusion models solve inverse problems accurately?

Diffusion models approximate the score of a target distribution to generate new samples. Here we ask how well a diffusion model learns the true data distribution's underlying features by posing this question as an inverse problem. When the target belongs to a parameterized family, can generated samples accurately identify the unknown parameter? We propose sensitivity of the diffusion score matching (DSM) loss as a mechanism for accurate parameter recovery. Specifically, we prove an information-theoretic sufficient condition under which generated samples recover a parameter accurately, even when the target score itself appears insensitive to parameter changes. For example, for Gaussian mixtures in arbitrary dimension, we show that the mixture-weight parameter is recoverable at the same order of accuracy as the DSM loss. This resolves an apparent paradox in the theoretical literature. Although the scores of well-separated Gaussian mixtures can yield poor mixture-weight estimates, we show that scores along the noising process remain sensitive to mixture weights and thereby enable accurate recovery. Experiments on benchmark datasets corroborate this mechanism: noise schedules designed to reduce sensitivity also degrade parameter recovery. Our results characterize when diffusion models can capture rare modes in multimodal mixtures and, more broadly, when diffusion-generated samples can support accurate solutions to scientific inverse problems.