Diffusion with Forward Models: Solving Stochastic Inverse Problems Without Direct Supervision

Proposition 1. Suppose that any signal The total observation loss is defined in Equation equation 4 below. After introducing some notation, we will formalize the assumptions made in the proposition. Definition 2. We define the scattering map as the (measurable) map sending signal In other words, given all possible observations of a signal, we can uniquely reconstruct the signal (for the class of signals under consideration). Observations generated by our model are slices of total observations. Thus, our model is limited to modeling the space over observations that are a member of the total observations set, i.e., The predicted distribution over signals can be recovered from the distribution over observations.