Rapid Grassmannian Averaging with Chebyshev Polynomials

TL;DR

This paper introduces fast algorithms for averaging points on Grassmannian manifolds, significantly reducing computational complexity and enabling efficient applications in machine learning, computer vision, and signal processing.

Contribution

The paper presents novel spectral-based algorithms, RGrAv and DRGrAv, for rapid centralized and decentralized Grassmannian averaging, with theoretical guarantees and superior empirical performance.

Findings

Algorithms outperform existing methods in speed and accuracy.

The methods are effective in tasks like K-means clustering on video data.

The algorithms are versatile for various Grassmannian averaging applications.

Abstract

We propose new algorithms to efficiently average a collection of points on a Grassmannian manifold in both the centralized and decentralized settings. Grassmannian points are used ubiquitously in machine learning, computer vision, and signal processing to represent data through (often low-dimensional) subspaces. While averaging these points is crucial to many tasks (especially in the decentralized setting), existing methods unfortunately remain computationally expensive due to the non-Euclidean geometry of the manifold. Our proposed algorithms, Rapid Grassmannian Averaging (RGrAv) and Decentralized Rapid Grassmannian Averaging (DRGrAv), overcome this challenge by leveraging the spectral structure of the problem to rapidly compute an average using only small matrix multiplications and QR factorizations. We provide a theoretical guarantee of optimality and present numerical experiments which demonstrate that our algorithms outperform state-of-the-art methods in providing high accuracy solutions in minimal time. Additional experiments showcase the versatility of our algorithms to tasks such as K-means clustering on video motion data, establishing RGrAv and DRGrAv as powerful tools for generic Grassmannian averaging.