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Manifold learning and optimization using tangent space proxies

Ryan A. Robinett, Lorenzo Orecchia, Samantha J. Riesenfeld

TL;DR

This work proposes an atlas-graph framework for efficient differential-geometric computations on arbitrary manifolds, enabling fast first-order optimization on the Grassmann manifold and the learning of tangent-space proxies from point clouds such as high-contrast image patches. By constructing an atlas of overlapping coordinate charts, the method accelerates optimization and facilitates downstream learning tasks, including a Riemannian SVM that leverages learned geometry to approximate logarithmic maps and vector transports. The approach extends to high-dimensional, noisy ambient spaces, offering a practical pathway to integrating differential-geometric primitives into scalable machine learning pipelines. Overall, the atlas-graph perspective provides a principled, data-driven way to harness manifold structure for both optimization and learning on complex geometric data.

Abstract

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.

Manifold learning and optimization using tangent space proxies

TL;DR

This work proposes an atlas-graph framework for efficient differential-geometric computations on arbitrary manifolds, enabling fast first-order optimization on the Grassmann manifold and the learning of tangent-space proxies from point clouds such as high-contrast image patches. By constructing an atlas of overlapping coordinate charts, the method accelerates optimization and facilitates downstream learning tasks, including a Riemannian SVM that leverages learned geometry to approximate logarithmic maps and vector transports. The approach extends to high-dimensional, noisy ambient spaces, offering a practical pathway to integrating differential-geometric primitives into scalable machine learning pipelines. Overall, the atlas-graph perspective provides a principled, data-driven way to harness manifold structure for both optimization and learning on complex geometric data.

Abstract

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.
Paper Structure (29 sections, 2 theorems, 7 equations, 2 figures, 2 tables, 1 algorithm)

This paper contains 29 sections, 2 theorems, 7 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Class options
  3. Front matter
  4. Cross references and hyperlinks
  5. Cleveref
  6. Hyperef
  7. Math and equations
  8. Theorem-like environments
  9. Tables
  10. Figures
  11. Algorithms
  12. Sections
  13. Supplemental material
  14. Template
  15. Bibliography
  16. ...and 14 more sections

Key Result

Theorem 6.1

\newlabelthm:mvt0 Suppose $f$ is a function that is continuous on the closed interval $[a,b]$. and differentiable on the open interval $(a,b)$. Then there exists a number $c$ such that $a < c < b$ and In other words, $f(b)-f(a) = f'(c)(b-a)$.

Figures (2)

  • Figure 1: Example figure using external image files.
  • Figure 2: Example PGFPLOTS figure.

Theorems & Definitions (5)

  • Theorem 6.1: Mean Value Theorem
  • Corollary 6.2
  • Proof 1
  • Claim 6.3
  • Proof 2: Proof of main theorem