  The scope of this paper is generative modeling through diffusion processes.
An approach falling within this paradigm is the work of Song et al. (2021),
which relies on a time-reversal argument to construct a diffusion process
targeting the desired data distribution. We show that the time-reversal
argument, common to all denoising diffusion probabilistic modeling proposals,
is not necessary. We obtain diffusion processes targeting the desired data
distribution by taking appropriate mixtures of diffusion bridges. The resulting
transport is exact by construction, allows for greater flexibility in choosing
the dynamics of the underlying diffusion, and can be approximated by means of a
neural network via novel training objectives. We develop a unifying view of the
drift adjustments corresponding to our and to time-reversal approaches and make
use of this representation to inspect the inner workings of diffusion-based
generative models. Finally, we leverage on scalable simulation and inference
techniques common in spatial statistics to move beyond fully factorial
distributions in the underlying diffusion dynamics. The methodological advances
contained in this work contribute toward establishing a general framework for
generative modeling based on diffusion processes.
