{"slug": "diffeomorphic-optimization", "title": "Diffeomorphic Optimization", "summary": "A paper submitted to arXiv on 1 July 2026 introduces diffeomorphic optimization, a method that uses diffusion and flow models to map the data manifold to a simpler base space and perform gradient descent there, which the authors show is equivalent to Riemannian gradient descent on the data manifold up to O(λ²) corrections. In protein design, the method places 91.3% of residues in the Ramachandran target with FrameFlow versus 63.3% for tuned guidance, 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. The work also derives an autograd-compatible SO(3) gradient and a generalized adjoint-state method for backpropagation through Lie-group ODE solvers.", "body_md": "# Computer Science > Machine Learning\n\n  [Submitted on 1 Jul 2026]\n\n# Title:Diffeomorphic Optimization\n\n[View PDF](https://arxiv.org/pdf/2607.00947)\n\n[HTML (experimental)](https://arxiv.org/html/2607.00947v1)\n\nAbstract: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 $\\mathcal{O}(\\lambda^2)$ 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.\n    \n\n### Additional Features\n\n### References & Citations\n\nLoading...\n\n# Bibliographic and Citation Tools\n\nBibliographic Explorer \n\n*(*[What is the Explorer?](https://info.arxiv.org/labs/showcase.html#arxiv-bibliographic-explorer))\nConnected Papers \n\n*(*[What is Connected Papers?](https://www.connectedpapers.com/about))\nLitmaps \n\n*(*[What is Litmaps?](https://www.litmaps.co/))\nscite Smart Citations \n\n*(*[What are Smart Citations?](https://www.scite.ai/))\n# Code, Data and Media Associated with this Article\n\nalphaXiv \n\n*(*[What is alphaXiv?](https://alphaxiv.org/))\nCatalyzeX Code Finder for Papers \n\n*(*[What is CatalyzeX?](https://www.catalyzex.com))\nDagsHub \n\n*(*[What is DagsHub?](https://dagshub.com/))\nGotit.pub \n\n*(*[What is GotitPub?](http://gotit.pub/faq))\nHugging Face \n\n*(*[What is Huggingface?](https://huggingface.co/huggingface))\nScienceCast \n\n*(*[What is ScienceCast?](https://sciencecast.org/welcome))\n# Demos\n\n# Recommenders and Search Tools\n\nInfluence Flower \n\n*(*[What are Influence Flowers?](https://influencemap.cmlab.dev/))\nCORE Recommender \n\n*(*[What is CORE?](https://core.ac.uk/services/recommender))\nIArxiv Recommender\n\n*(*[What is IArxiv?](https://iarxiv.org/about))\n# arXivLabs: experimental projects with community collaborators\n\narXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.\n\nBoth individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.\n\nHave an idea for a project that will add value for arXiv's community? [**Learn more about arXivLabs**](https://info.arxiv.org/labs/index.html).", "url": "https://wpnews.pro/news/diffeomorphic-optimization", "canonical_source": "https://arxiv.org/abs/2607.00947", "published_at": "2026-10-21 03:10:52+00:00", "updated_at": "2026-10-21 03:18:14.692462+00:00", "lang": "en", "topics": ["machine-learning", "ai-research", "generative-ai", "artificial-intelligence"], "entities": ["arXiv", "FrameFlow", "OC-Flow", "Rosetta", "PDB", "SO(3)", "SE(3)"], "also_reported_by": [], "alternates": {"html": "https://wpnews.pro/news/diffeomorphic-optimization", "markdown": "https://wpnews.pro/news/diffeomorphic-optimization.md", "text": "https://wpnews.pro/news/diffeomorphic-optimization.txt", "jsonld": "https://wpnews.pro/news/diffeomorphic-optimization.jsonld"}}