# Manifold learning and optimization using tangent space proxies

arXiv: https://arxiv.org/abs/2501.12678 · HTML: https://mlsanity.com/paper/2501.12678/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 50 references that are in the map.
- First posted on arXiv 22 Jan 2025.

## Builds on (most cited first, 11 of 11)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Parameterized quantum circuits as machine learning models](https://mlsanity.com/paper/1906.07682/) (2019)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation](https://mlsanity.com/paper/1603.03236/) (2016)
- [The non-convex Burer-Monteiro approach works on smooth semidefinite programs](https://mlsanity.com/paper/1606.04970/) (2016)
- [Riemannian Optimization via Frank-Wolfe Methods](https://mlsanity.com/paper/1710.10770/) (2017)
- [An Alternative to EM for Gaussian Mixture Models: Batch and Stochastic Riemannian Optimization](https://mlsanity.com/paper/1706.03267/) (2017)
- [Online Principal Component Analysis in High Dimension: Which Algorithm to Choose?](https://mlsanity.com/paper/1511.03688/) (2015)
- [On the Optimality of the Oja's Algorithm for Online PCA](https://mlsanity.com/paper/2104.00512/) (2021)
- [On the Burer-Monteiro method for general semidefinite programs](https://mlsanity.com/paper/1904.07147/) (2019)
- [Principal Boundary on Riemannian Manifolds](https://mlsanity.com/paper/1711.06705/) (2017)

## Built on by (most cited first, 2 of 2)

- [Iso-Riemannian Optimization on Learned Data Manifolds](https://mlsanity.com/paper/2510.21033/) (2025)
- [Riemannian Archetypal Analysis: Interpretable non-linear data analysis on deformed star distributions](https://mlsanity.com/paper/2605.24113/) (2026)

## Abstract (arXiv)

We present a framework for efficiently approximating differential-geometric primitives on arbitrary manifolds via construction of an atlas graph representation, which leverages the canonical characterization of a manifold as a finite collection, or atlas, of overlapping coordinate charts. We first show the utility of this framework in a setting where the manifold is expressed in closed form, specifically, a runtime advantage, compared with state-of-the-art approaches, for first-order optimization over the Grassmann manifold. Moreover, using point cloud data for which a complex manifold structure was previously established, i.e., high-contrast image patches, we show that an atlas graph with the correct geometry can be directly learned from the point cloud. Finally, we demonstrate that learning an atlas graph enables downstream key machine learning tasks. In particular, we implement a Riemannian generalization of support vector machines that uses the learned atlas graph to approximate complex differential-geometric primitives, including Riemannian logarithms and vector transports. These settings suggest the potential of this framework for even more complex settings, where ambient dimension and noise levels may be much higher.