Manifold learning and optimization using tangent space proxies

TL;DR

This paper introduces a framework for approximating differential-geometric primitives on manifolds using an atlas graph, enabling efficient optimization and machine learning tasks on complex manifold data.

Contribution

The authors propose a novel atlas graph framework that learns manifold geometry from point clouds and improves optimization and learning algorithms on manifolds.

Findings

Faster first-order optimization on Grassmann manifolds.

Successful learning of manifold structure from high-contrast image patches.

Effective Riemannian SVM using the learned atlas graph.

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.