Diffeomorphic Optimization

Generative models learn data distributions that reside on a low-dimensional manifold within a higher-dimensional ambient space. Optimizing differentiable objectives on this manifold is challenging: the ambient loss landscape is high-dimensional, rugged, and non-convex. Direct gradient descent, blind to the manifold's geometry, quickly drifts off it. Diffeomorphic optimization starts from the observation that diffusion and flow models provide a map from the data manifold to a much simpler base space in which we perform gradient descent. Using differential geometry, we show this is equivalent to Riemannian gradient descent on the data manifold up to O(λ2) corrections, keeping trajectories on-manifold by construction and yielding a smoother optimization surface. For protein design, we extend diffeomorphic optimization to the matrix Lie groups SO(3) and SE(3), deriving an autograd-compatible SO(3) gradient and a generalized adjoint-state method for backpropagation through Lie-group ODE solvers. Diffeomorphic optimization improves over tuned guidance on secondary-structure targeting with FrameFlow (91.3% vs. 63.3% of residues in the Ramachandran target), outperforms OC-Flow on peptide binding affinity at 2× the speed, and reduces Rosetta energies by thousands of units across the PDB test set for structures with hundreds of residues.