{"ID":5551658,"CreatedAt":"2026-07-02T01:54:51.863792489Z","UpdatedAt":"2026-07-04T13:37:00.247962456Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.00947","arxiv_id":"2607.00947","title":"Diffeomorphic Optimization","abstract":"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 $\\mathcal{O}(λ^2)$ 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 $\\mathrm{SO}(3)$ and $\\mathrm{SE}(3)$, deriving an autograd-compatible $\\mathrm{SO}(3)$ 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 $2\\times$ the speed, and reduces Rosetta energies by thousands of units across the PDB test set for structures with hundreds of residues.","short_abstract":"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 g...","url_abs":"https://arxiv.org/abs/2607.00947","url_pdf":"https://arxiv.org/pdf/2607.00947v1","authors":"[\"Ludwig Winkler\",\"Andrew Leaver-Fay\",\"Joseph Kleinhenz\",\"Pan Kessel\"]","published":"2026-07-01T13:46:22Z","proceeding":"cs.LG","tasks":"[\"cs.LG\"]","methods":"[\"Diffusion Model\"]","has_code":false}