  Diffusion models excel at creating visually-convincing images, but they often
struggle to meet subtle constraints inherent in the training data. Such
constraints could be physics-based (e.g., satisfying a PDE), geometric (e.g.,
respecting symmetry), or semantic (e.g., including a particular number of
objects). When the training data all satisfy a certain constraint, enforcing
this constraint on a diffusion model not only improves its
distribution-matching accuracy but also makes it more reliable for generating
valid synthetic data and solving constrained inverse problems. However,
existing methods for constrained diffusion models are inflexible with different
types of constraints. Recent work proposed to learn mirror diffusion models
(MDMs) in an unconstrained space defined by a mirror map and to impose the
constraint with an inverse mirror map, but analytical mirror maps are
challenging to derive for complex constraints. We propose neural approximate
mirror maps (NAMMs) for general constraints. Our approach only requires a
differentiable distance function from the constraint set. We learn an
approximate mirror map that pushes data into an unconstrained space and a
corresponding approximate inverse that maps data back to the constraint set. A
generative model, such as an MDM, can then be trained in the learned mirror
space and its samples restored to the constraint set by the inverse map. We
validate our approach on a variety of constraints, showing that compared to an
unconstrained diffusion model, a NAMM-based MDM substantially improves
constraint satisfaction. We also demonstrate how existing diffusion-based
inverse-problem solvers can be easily applied in the learned mirror space to
solve constrained inverse problems.
