# Iso-Riemannian Optimization on Learned Data Manifolds arXiv: https://arxiv.org/abs/2510.21033 ยท HTML: https://mlsanity.com/paper/2510.21033/ ## Key facts - Cited by 4 papers in the map (data through 14 Oct 2026). - Builds on 34 of its 66 references that are in the map. - First posted on arXiv 23 Oct 2025. ## Builds on (most cited first, 30 of 34) - [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018) - [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014) - [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021) - [Phase Retrieval Under a Generative Prior](https://mlsanity.com/paper/1807.04261/) (2018) - [Intermediate Layer Optimization for Inverse Problems using Deep Generative Models](https://mlsanity.com/paper/2102.07364/) (2021) - [Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections](https://mlsanity.com/paper/2404.02954/) (2024) - [A Geometric View of Data Complexity: Efficient Local Intrinsic Dimension Estimation with Diffusion Models](https://mlsanity.com/paper/2406.03537/) (2024) - [Global Guarantees for Enforcing Deep Generative Priors by Empirical Risk](https://mlsanity.com/paper/1705.07576/) (2017) - [Your diffusion model secretly knows the dimension of the data manifold](https://mlsanity.com/paper/2212.12611/) (2022) - [Fast and accurate optimization on the orthogonal manifold without retraction](https://mlsanity.com/paper/2102.07432/) (2021) - [Inverting Deep Generative models, One layer at a time](https://mlsanity.com/paper/1906.07437/) (2019) - [Optimization Techniques on Riemannian Manifolds](https://mlsanity.com/paper/1407.5965/) (2014) - [A Locally Adaptive Normal Distribution](https://mlsanity.com/paper/1606.02518/) (2016) - [Riemannian Optimization via Frank-Wolfe Methods](https://mlsanity.com/paper/1710.10770/) (2017) - [Riemannian Metric Learning via Optimal Transport](https://mlsanity.com/paper/2205.09244/) (2022) - [Infeasible Deterministic, Stochastic, and Variance-Reduction Algorithms for Optimization under Orthogonality Constraints](https://mlsanity.com/paper/2303.16510/) (2023) - [Adaptive regularization with cubics on manifolds](https://mlsanity.com/paper/1806.00065/) (2018) - [Simple algorithms for optimization on Riemannian manifolds with constraints](https://mlsanity.com/paper/1901.10000/) (2019) - [A Review on Riemannian Metric Learning: Closer to You than You Imagine](https://mlsanity.com/paper/2503.05321/) (2025) - [Manifold Approximation by Moving Least-Squares Projection (MMLS)](https://mlsanity.com/paper/1606.07104/) (2016) - [Learning Distances from Data with Normalizing Flows and Score Matching](https://mlsanity.com/paper/2407.09297/) (2024) - [Pulling back symmetric Riemannian geometry for data analysis](https://mlsanity.com/paper/2403.06612/) (2024) - [Optimization without Retraction on the Random Generalized Stiefel Manifold](https://mlsanity.com/paper/2405.01702/) (2024) - [Optimal Regularization for a Data Source](https://mlsanity.com/paper/2212.13597/) (2022) - [Manifold Learning with Normalizing Flows: Towards Regularity, Expressivity and Iso-Riemannian Geometry](https://mlsanity.com/paper/2505.08087/) (2025) - [The difference of convex algorithm on Hadamard manifolds](https://mlsanity.com/paper/2112.05250/) (2021) - [Manifold Free Riemannian Optimization](https://mlsanity.com/paper/2209.03269/) (2022) - [Riemannian Interior Point Methods for Constrained Optimization on Manifolds](https://mlsanity.com/paper/2203.09762/) (2022) - [Fenchel Duality Theory and A Primal-Dual Algorithm on Riemannian Manifolds](https://mlsanity.com/paper/1908.02022/) (2019) - [An Inexact Semi-smooth Newton Method on Riemannian Manifolds with Application to Duality-based Total Variation Denoising](https://mlsanity.com/paper/2102.10309/) (2021) ## Built on by (most cited first, 4 of 4) - [Riemannian Metric Matching for Scalable Geometric Modeling of Distributions](https://mlsanity.com/paper/2606.14334/) (2026) - [Riemannian AmbientFlow: Towards Simultaneous Manifold Learning and Generative Modeling from Corrupted Data](https://mlsanity.com/paper/2601.18728/) (2026) - [Riemannian Archetypal Analysis: Interpretable non-linear data analysis on deformed star distributions](https://mlsanity.com/paper/2605.24113/) (2026) - [Beyond Unimodal Bases: Pullback Geometry for Multimodal Data](https://mlsanity.com/paper/2610.00708/) (2026) ## Abstract (arXiv) 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.