Grassmannian Learning: Embedding Geometry Awareness in Shallow and Deep Learning

TL;DR

This paper introduces Grassmannian learning, highlighting its mathematical foundations and applications in signal processing, and discusses how it enhances both shallow and deep learning models.

Contribution

It provides a comprehensive survey of Grassmannian learning, covering mathematical problems, solution approaches, and diverse applications, promoting its adoption in various fields.

Findings

Grassmannian learning improves performance in signal-processing tasks.

Both shallow and deep models benefit from Grassmann manifold techniques.

The survey encourages wider adoption of Grassmannian methods.

Abstract

Modern machine learning algorithms have been adopted in a range of signal-processing applications spanning computer vision, natural language processing, and artificial intelligence. Many relevant problems involve subspace-structured features, orthogonality constrained or low-rank constrained objective functions, or subspace distances. These mathematical characteristics are expressed naturally using the Grassmann manifold. Unfortunately, this fact is not yet explored in many traditional learning algorithms. In the last few years, there have been growing interests in studying Grassmann manifold to tackle new learning problems. Such attempts have been reassured by substantial performance improvements in both classic learning and learning using deep neural networks. We term the former as shallow and the latter deep Grassmannian learning. The aim of this paper is to introduce the emerging area of Grassmannian learning by surveying common mathematical problems and primary solution approaches, and overviewing various applications. We hope to inspire practitioners in different fields to adopt the powerful tool of Grassmannian learning in their research.