Iso-Riemannian Optimization on Learned Data Manifolds

[2510.21033] Iso-Riemannian Optimization on Learned Data Manifolds Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Optimization and Control arXiv:2510.21033 (math) [Submitted on 23 Oct 2025 ( v1 ), last revised 14 Aug 2026 (this version, v3)] Title: Iso-Riemannian Optimization on Learned Data Manifolds Authors: Willem Diepeveen , Melanie Weber View a PDF of the paper titled Iso-Riemannian Optimization on Learned Data Manifolds, by Willem Diepeveen and 1 other authors View PDF HTML (experimental) Abstract: We develop a theory of iso-Riemannian optimization for problems constrained to learned data manifolds, a setting in which classical Riemannian optimization - and Riemannian gradient descent in particular - can be poorly suited. That is, favorable Euclidean properties of an objective need not translate into geodesic convexity or L-smoothness, and the Riemannian gradient may provide an unsuitable search direction. We instead combine the manifold mappings induced by the iso-connection, whose geodesics have constant Euclidean speed, with the l2-projected Euclidean gradient, resulting in l2-projected gradient iso-Riemannian descent, in an attempt to alleviate these issues. We analyze this scheme from two complementary perspectives. First, we introduce iso-g-convexity and iso-L-smoothness, relate strong iso-g-convexity to a Polyak-Lojasiewicz-type condition, and establish general convergence guarantees. This function-based theory addresses the conditioning and search-direction issues that motivate our approach. Second, we introduce iso-monotonicity and iso-Lipschitzness for vector fields. However, while these notions yield analogous convergence results in one dimension and enable applications such as the computation of iso-Riemannian barycentres, the resulting theory need not extend meaningfully to higher dimensions. So unlike in the classical Levi-Civita setting, these function- and vector field-based perspectives need not be equivalent in the iso-Riemannian setting. Consequently, distinct extensions of classical Riemannian optimization lead to different assumptions and convergence guarantees. The theory developed here suggests that, for function optimization, the function-based perspective provides the more general, tractable and overall suitable framework, whereas the vector field perspective is naturally reserved for problems without a direct function-based formulation. Subjects: Optimization and Control (math.OC) ; Machine Learning (cs.LG); Differential Geometry (math.DG) MSC classes: 90C26, 68T07, 53Z50 Cite as: arXiv:2510.21033 [math.OC] (or arXiv:2510.21033v3 [math.OC] for this version) https://doi.org/10.48550/arXiv.2510.21033 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Willem Diepeveen [ view email ] [v1] Thu, 23 Oct 2025 22:34:55 UTC (1,249 KB) [v2] Wed, 6 May 2026 17:29:12 UTC (1,265 KB) [v3] Fri, 14 Aug 2026 22:41:31 UTC (1,975 KB) Full-text links: Access Paper: View a PDF of the paper titled Iso-Riemannian Optimization on Learned Data Manifolds, by Willem Diepeveen and 1 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.OC < prev | next > new | recent | 2025-10 Change to browse by: cs cs.LG math math.DG References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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