Nystr\"om Approximation on Manifolds

TL;DR

This paper introduces a Riemannian Nyström approximation method for tangent operators on manifolds, reducing computational costs while preserving key properties, and applies it to optimization and data analysis tasks.

Contribution

It develops a low-rank, intrinsic approximation technique on manifolds using Haar--Grassmann sketching, enhancing efficiency in manifold computations.

Findings

Reduces computational costs of tangent operators on manifolds.

Maintains positive semidefiniteness and approximation accuracy.

Effective in principal geodesic analysis on real data.

Abstract

Computations on a manifold often involve constructing an operator on the tangent space and computing its inverse, which can be time-consuming in many applications. In order to reduce the computational costs and preserve the benign properties of tangent operators, we develop the Riemannian Nystr\"om approximation on manifolds, a low-rank approximation of tangent operators through subspace projections onto the tangent space. The developed approximation is intrinsically constructed and inherits desirable properties from the classical Nystr\"om approximation, e.g., positive semidefiniteness and approximation errors. Instead of the Gaussian sketching, we introduce the Haar--Grassmann sketching condition with a coordinate-free representation, which remains compatible under isometric vector transport across tangent spaces. Moreover, we propose a randomized Newton-type method for optimization on manifolds in which the linear system is constructed via the Riemannian Nystr\"om approximation. Numerical experiments on the SPD and Grassmann manifolds, together with principal geodesic analysis on real data, illustrate that the proposed approximation reduces the computational cost of operators while maintaining comparable accuracy.