{"id":1178886,"date":"2026-07-16T10:12:20","date_gmt":"2026-07-16T17:12:20","guid":{"rendered":"https:\/\/find.codeghost.online\/en-us\/research\/publication\/diffeomorphic-optimization\/"},"modified":"2026-07-19T15:53:48","modified_gmt":"2026-07-19T22:53:48","slug":"diffeomorphic-optimization","status":"publish","type":"msr-research-item","link":"https:\/\/find.codeghost.online\/en-us\/research\/publication\/diffeomorphic-optimization\/","title":{"rendered":"Diffeomorphic Optimization"},"content":{"rendered":"\n\n\n<p class=\"wp-block-paragraph\">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&#8217;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 <math><mrow><mi>\ud835\udcaa<\/mi><mo stretchy=\"false\">(<\/mo><msup><mi>\u03bb<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><\/mrow><\/math> 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 <math><mrow><mtext>SO<\/mtext><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/math> and <math><mrow><mtext>SE<\/mtext><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/math>, deriving an autograd-compatible <math><mrow><mtext>SO<\/mtext><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/math> 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 (<math><semantics><mrow><mtext>91.3%<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">91.3%<\/annotation><\/semantics><\/math> vs. <math><semantics><mrow><mtext>63.3%<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">63.3%<\/annotation><\/semantics><\/math> of residues in the Ramachandran target), outperforms OC-Flow on peptide binding affinity at <math><mrow><mn>2<\/mn><mo>\u00d7<\/mo><\/mrow><\/math> the speed, and reduces Rosetta energies by thousands of units across the PDB test set for structures with hundreds of residues.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>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&#8217;s geometry, quickly drifts off it. Diffeomorphic optimization starts from the observation that diffusion and flow [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":[{"type":"user_nicename","value":"Ludwig Winkler","user_id":"43976"},{"type":"text","value":"Andrew Leaver-Fay","user_id":0},{"type":"text","value":"J. 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